XIV. SATURATION OF NUCLEAR FORCES The binding energy and volume of nuclei are proportional to A. the mass number. This is not in accord a villi a, law of force which gives equal interactions between all pairs of particles in the nucleus for there are then A. [A — l)/2 distinct interacting pairs and a binding energy at least proportional to A (A — l)/2 might be expected, if not to a higher power of A. due to increased packing with more interaction. Instead, the nuclear binding energies seem similar to the internal energies of bulk matter, in which 2 pounds has twice as much energy and volume as 1 pound. To account for this phenomenon of "saturation of nuclear forces," in which one particle apparently interacts with only a limited number of others, various hypotheses have been made, and various other assumptions about the nature of the forces can be shown to bo impossible. Among the impossible assumptions is that which has been used in this book so far, namely, an ordinary potential independent of the angular momentum, because it is easily shown that such a potential does not give saturation. This is so even if the Coulomb repulsion of the protons is taken into account. The proof can be carried out with various degrees of exactness, using the variational method. This method is based ou the Schrodinger variational theorem which states that the quantity 9. = $f1B$ dr/jy dr (1-14) is a minimum when ^ is the correct eigenf unction of the lowest eigenvalue E (i of //, and the minimum value of Q is E$. Thus, it the assumed Hamil Ionian operator representing the interaction of the particles in a given nucleus is sandwiched between any arbitrary ip in the expression for 11, the value of must be greater (i.e., less negative) than the correct energy of that nucleus. The simplest i^'s are plane waves inside a box representing the nucleus. If the size of the box is adjusted to give as low an as possible, this size comes out about equal to the range of nuclear forces, which is clearly much too small. Further, it gives a potcntia 80 SATURATION OF NUCLEAR FORCES SI nergy proportional to A 2 , and a kinetic energy proportional to a# The size of the coefficients of these powers is such that the J otential energy dominates for A > 50 ; for .4 = 238 the binding energy is greater than 238 mass units. This is convincing evidence; that the ordinary potential will not work, and this is true inde- pendently of the shape of the potential (square well, exponential, Gaussian, etc.). What is needed is a potential which prevents the particles from getting too close together. A. poten fial repulsive at short distances, originally used by Morse for molecules, has been explored by Schiff and Fisk; the only objection is that the high repulsive potential may give relativistic difficulties if it gets above 2Mc 2 ~ 1800 Mev, for a proton in such a state would have nega- tive kinetic energy. However, the idea of a repulsive potential has not been followed up sufficiently. EXCHANGE FORCES In the first paper on nuclear forces, TTeisenberg proposed, in order to explain the saturation of nuclear forces, that these forces are "exchanger" forces, similar to the force that binds ordinary chemical molecules. Without inquiring into the origin of these exchange forces, let us write down the various types of exchange forces that can exist between two particles, and then examine the effects of these forces on the properties of the deuteron, and on the saturation of the binding energy. For an ordinary (non-exchange) central force the Schrodinger equation for two particles is (in the center-of-mass system) : [(n 2 /M)V 3 + Etyixu % frj, ffjO 7(^(^*3* <rt„vz} (W5) Wigner In nuclear physics, such forces are called Wigner forces. The interaction does not cause any exchange between, coordinates of the two particles. Another type of interaction is bile that inter- changes the space coordinates of the two particles in addition to multiplication of $ by some V(r); for such an interaction, the Schrodinger equation is : [{ii 2 /M)V 2 + Emt h t,, ffu ad = FWtffct, r,, <r u *a) (146) Major ana XIV. SATURATION OF NUCLEAR FORCES The binding energy and volume of nuclei are proportional to A, the mass number. This is not in accord with a law of force which gives equal interactions between all pairs of particles in the nucleus, for there are then A (A — J.)/2 distinct interacting pairs and a binding energy at least proportional to A(A — L)/2 might be expected, if not to a higher power of .4 clue to increased packing with more interaction. Instead, the nuclear binding energies seem similar to the internal energies of bulk matter, in which 2 pounds has twice as much energy and volume as 1 pound. To account for this phenomenon of ''saturation of nuclear forces," in which one particle apparently interacts wdth only a limited number of others, various hypotheses have been made, and various other assumptions about the nature of the forces can be shown to be impossible. Among the impossible assumptions is that which has been used in this book so far, namely, an ordinary potential independent of the angular momentum, because ft is easily shown that such a potential does not give saturation. This is so even if the Coulomb repulsion of the protons is taken into account. The proof can be carried out "with various degrees of exactness, using the variational method. This method is based on the Schro dinger variational theorem which states that the quantity 9. = f$£$ dr/jf 2 dr (1443 is a minimum when ^ is the correct eigenfunetion of the lowest eigenvalue Eq of H, and the minimum value of V. is E$, Thus, if {hg assumed Ilamiltonian operator representing the interaction of the particles in a given nucleus is sandwiched between any arbitrary $ in the expression for fi, the value of must be greater (i.e., less negative) than the correct energy of that nucleus. The simplest ^'s are plane waves inside a box representing the nucleus, ff the size of the box is adjusted to give as low an 9. as possible, this size comes out about equal to the range; of nuclear forces, which is clearly much too small. Further, it gives a potential 80 SATURATION OF NUCLEAR FORCES SI energy proportional to A 2 , and a kinetic energy proportional to A H , The size of the coefficients of these powers is such that the potential energy dominates for A > 50; for A = 238 the binding energy is greater than 23S mass mats, This is convincing evidence that the ordinary potential will not work, and this is true inde- pendently of the shape of the potential (square well, exponential, Gaussian, etc.). What is needed is a potential which prevents the particles from getting too close together. A potential repulsive at short distances, originally used by Morse for molecules, has bees explored by Schiff and Fisk; the only objection is that the high repulsive potential may give relativlstic difficulties if it gets above 2il/c 2 -^ 1800 Mev, for a proton in such a state would have nega- tive kinetic energy. However, the idea of a repulsive potential has not been followed up sufficiently. EXCHANGE FORCES In the first paper on nuclear forces, ITeisenberg proposed, in order to explain the saturation of nuclear forces, that these forces are ''exchange" forces, similar to the force that binds ordinary chemical molecules. Without inquiring into the origin of these exchange forces, let us write down the various types of exchange forces that can exist between two particles, and then examine the effects of these forces on the properties of the deuteron, and on the saturation of the binding energy, For an ordinary (non-exchange) central force the Schrodinger equation for two particles is (in the center-of-mass system) : [(h 2 /M)V 2 + Ety(r u r 2 , a u <r. 2 ) Wigner In nuclear physics, such forces are called Wigner forces. The interaction does not cause any exchange between coordinates of the two particles. Another type of interaction is one that inter- changes the space coordinates of the two particles in addition to multiplication of # by some V(r); for such an in ferae tiou, the Schrodinger equation is: [(n 2 /M)V- + Ety(r h Z& « u <r 3 ) = V(r)f(i-2, *i, in,*& (146) Major ana S2 QUANTITATIVE THEORY OF NUCLEAR FORCES Such a force is called a Majorana force. Two other possibilities are: (1) the Bartlett force, with interchange of spin coordinates, and (2) the Helsenberg force, with interchange of both space and spin coordinates. The Schrodinger equations are respectively: [(h 3 /'M)¥ 2 + E\^(r u t 2 , ft, a,) = 7(r)^(r 1; r 3 , *% ft) (147) Bartlett [(hVAf)V 2 + m(r h r 2 , ft, cj) = V(r)#(t2 t r,, <r 2 , ft) (14S) ff&senherg Effects of Exchange Forces. Exchange forces, with a V(r), are central forces and do not cause mixing of I's. However, if a tensor force is used instead of V(r) as the multiplying potential, I's are mixed and the quadrapole moment of the deuieron may be ex- plained as before. It should be pointed out that the tensor force does not by itself lead to saturation; this was proved by Volkolf (Phys. Rev. 62, 134). Majorana Force. The Majorana interaction replaces (r) by (— r) in •■]/-. Using the well-known behavior of the wave function on such an inversion, the Schrodinger equation (146) may be rewritten UhVAOV 2 + EU® = (-l) ! 7(r)^(r) (149) This is equivalent to having an ordinary potential that changes sign according' to whether I is even or odd, and is independent of spin. Since the experimental data discussed so far give informa- tion on the potential only for 1= % we have as yet no direct evi- dence as to wh.et.her the potential is "ordinary" or of the Majorana type. Since the potential is attractive for I = 0, it would be equally repulsive for I = 1 M the interaction, were totally of the Majorana type. Bartlett Force. Considering still a system of two particles, the spin function is symmetric if the total spin B is 1, and antisym- metric if the total spin is 0. Thus, the Schrodinger equation (147) for the Bartlett force may be rewritten;; [(h a /w 2 + mm = oo^'f-ww $s® This is equivalent to an ordinary potential which changes sign between S = and £ = 1. Since we know from neutron-proton scattering data that both the :i S and *M potentials are attractive, the nuclear force cannot be totally of the Bartlett type. SATURATION OF NUCLEAR FORCES 83 Heisenherg Force. Combining the arguments of the two last paragraphs,' the Schrodinger equation (148) may be rewritten for the Heisenbcrg force: KhViiov 2 + Mum = c - iy +s+v mf(j) (m) This is equivalent to an ordinary potential which changes sign according to whether l+'$ is even or odd. For example:, the effective potential is: for 3 5' potential + V(r) ■v(f) Sp -Y(r) + 70') (152) The reversal of sign, between 3 $- and ^-states indicates, as for flu > Bartlett. force, that the nuclear force cannot he wholly of the Heisenherg type. However, the difference between the S S and *£ neutron-proton well depths (about 21 and 12 Mev, respectively, f or a _ 2.8 X I0~ ri cm) can be explained by assuming that the interaction is about 25 per cent Ileisenberg or Bartlett and 75 per cent Wigner or Majorana. Exchange Forces and Saturation. The Bartlett spin-exchange force does not lead to saturation of the binding energy per particle. If the nuclear force were of the Bartlett type, heavy nuclei should exist with all spins aligned where the number of interacting pairs is A (A - l)/2, which leads to binding energy proportional to at least the square of A. However, the space exchange in the Majorana and the Ileisen- berg forces does lead to saturation because of the alternation in sign of the potential between odd and even I. For example, assume the nuclear force is the Majorana type (we already know it cannot bo more than about 25 per cent Heisenherg). Then saturation should not be apparent in nuclei up to He 4 , for in He 4 the spatial wave function can still be symmetrical in all four particles, without violating the Pauli principle. We need only give antiparallel spins (antisymmetric spin wave functions) to the two n.iutrons, and likewise to the two protons. Thus the Majorana force does uot alter the Wigner argument about the short range of the forces based on the binding energies of He' 1 and lighter nuclei. In the next heavier nucleus— I Ie B or hi 5 — the Pauli principle can no longer be satisfied by spin wave functions alone; there- fore, the spatial wave function must have at least one node. In S-i QUANTITATIVE THEORY OF NUCLEAR FORCES other words, only four particles can be in an ,s-state, whereas the last has to be put in a potato, and will therefore be repelled by the other particles. He u and Li 5 should thus be unstable, in agreement with experiment. This is a first sign of saturation. To investigate saturation in heavy nuclei, one may use the same variational method used at the beginning of the present- chapter to prove that ordinary forces do not give saturation. It i.s satisfactory that this calculation, in the case of the Majorana force, does not lead to non-saturation. On the other hand, since the variational method gives only a maximum to the true energy, it cannot be used to prove that the Majorana force does give saturation. Bat Wigncr ha.s given a conclusive argument that saturation is achieved with the space-exchange Majorana force (Proc. N T at. Acad. Set. 22, 662, .1936). The space-exchange part of the Heisenberg force would also cause saturation. SPIN AND ISOTOPIC SPIN It is often convenient to write exchange forces in a slightly different way . Sin c e f o i ■ t w o par tides 0"i ■ o% = +1 for S = 1 = -3 for S = 0, (153) the Bartlett force between two particles can obviously be written as [+v¥&s- i -V(t), s = W®(X + *i ■ <%) (154) The spin-exchange part of the Heisenberg force; could be written in the same way. In order to be able to use a similar notation for the space- exchange part of forces, we introduce the concept of the charge of a particle as a coordinate, i.e., neutron and proton are regarded as different eigenstates of the same particle, called a nudeon. We choose the symbol t for this charge coordinate and we define My = }4 for the proton M T m — ]/i for the neutron T = Y % for both (1-55) SATURATION OF NUCLEAR FORCES 85 using + H in analogy with the spin coordinate. We also define; the charge functions Charge function = y for the proton = 5 for the neutron (156) in analogy with the spin functions a and 8. The nucleons must obey Fermi statistics in order to be con- sistent- with the ordinary theory (this will become apparent shortly, if it is not immediately obvious). Thus the total wave function (including the charge function) for two or more particles 4> = #spafiB&3 'AspmW vWrKeO) (157) must be antisymmetric with respect to inter change of ail coordi- nates of two iiueleons. We therefore look for symmetric and anti- symmetric charge functions for two particles. Then; are four of these, as given in Table 4. TABLE 4 Twq-Pa:rticle Charge Functions Rtspi'i.;- Net State Functiou seating Symmetry Charge I 7(l)v(2) He 2 symmetric 2e II 5(I)5( 2 ) ™ 2 symmetric III (1./v / 2)[t(1)5(2) +7(2)5(1)1 H- symmelEe e IV (l/v / 2)[7(l)5(2) — 7(2)5(1)1 IT- antisymmetric c Again, in analogy to spin, two quantum numbers are defined to describe these functions: T to describe symmetry, and M T to describe the net charge. These quantities have the values given in Table 5. TABLE 5 Quantum Nu-Mfstchs fok Chabge States Stale T M r I 1 I II 1 -1 in 1 IV W QUANTITATIVE THEORY OF NIT CLEAR FORCES T is 1 for symmetric functions, for the antisymmetric function, in analogy to spin. M r is the sum of the M r 's for the two nueleons. In the literature t is called the "isotopic spin," T is called "the total isotopic spin," and M T may be called the "component of r in the direction of positive charge." T is analogous to total spin S, and Mj to B z . Tor a given T, M T can have the values T, T — 1, -■■,-T. From Table 4 it is seen that a system containing two neutrons or two protons has a symmetric charge function. Since we arc assuming nueleons to obey Fermi statistics, the remainder of the wave function (147) must be antisymmetric. This implies (cor- rectly) Fermi statistics for neutrons and protons, disregarding charge as a coordinate. But in a system containing a neutron and a proton the charge function can be either symmetric or anti- symmetric, and so also can the remainder of the wave function. Therefore, the treatment of proton and neutron as two eigens fates of the same particle does not in this case introduce any restric- tions, consistent with the ordinary theory of statistics-. it; is also convenient to introduce an operator t in analogy to the cr operator, defined by its effect on the "charge coordinate" M T . The eigenvalue of its absolute square is, again in analogy with spin: |t| 2 - Vf(T + 1) (158) Then, just as for spin, in a system of two nucleons Ti -To = +1 for T = 1 -3 for T = (159) Now the Heisenberg interaction can be written (letting V(r) absorb the factor —1) as ¥>V{r){\ + T] -t.) (160) To prove this, we note that equation 100 changes sign according to whether the charge part of the wave function (equation 157) is symmetric or antisymmetric, i.e., according to whether the product of space and spin functions is antisymmetric or symmetric, which is just what is required according to equations 151 and 152. The types of interaction between the two particles discussed so far may now be summarized by listing the various types of oper- SATURATTON OF NUCLEAR FORCES 87 ators, which when multiplied by some V{r) give the interactions listed in Table 6. Eisenbud and Wigner (Proc. Nat. Acad, Sei. 27, 281) have shown that these interactions and their linear combinations are the onlv TABLE G Types of INTERACTIONS Ordinary Spin cxdiiinge Spaee^spin exulian^o Space exchange Tensor Tenser exchange 1 <H 0-2 Tl ■ T2 (<T1-0- 2 )(ti-to) (<ri-r)(o-2-r) (q-l ■ r) (0-2 ■ r) (Tl • Tn) Ones possible under certain reasonable invariance requirements, namely, excluding interactions depending on total eharge or on the momentum. (The in tea 'action (tri H- ov>) • L depends on the momentum.) QUANTITATIVE THEORY OF EXCHANGE FORCES In the last chapter, it was shown that the ground state of the deuteron, the neutron-proton scattering, and the quadrupole moment, of the deuteron could be obtained quantitatively by assuming a neutron-proton interaction of the form V (even) = -(1 - ¥& + US °T ' *z -I- ySrd.Kr) (161) with Jir) = V'o r < a J(r) = r > a g = 0.0715 V = 13.89 Mev 7 = 0.775 a = 2.80 X m~ 13 cm — Rarita anil Sch winger. The neutron-proton interaction (.161) applies only to states of L = 0. The potential for other L is as yet arbitrary. If wc assume in particular a fores of the type discussed in this chapter, i.e., depending only on the product of the isotopic spins X\ ■ r 2 , the potential will depend only on the parity of the state. The poten- tial for states of odd parity can only be determined from that for 88 QUANTITATIVE THEORY OF NUCLEAR FORCES slates of even parity by making some assumption regarding the exchange character (or dependence on Tj ■ fa) of the forces. Rarita and Sch winger chose; to investigate three potentials which were suggested by three types of meson theory (see Chapter XV) : I. Symmetric meson theory. II. Exchange forces, or charged meson theory. III. Ordinary forces, or neutral meson theory. These potentials are: I. II. III. " = — }ifi • t 2 o-i 0"2 V&yim [■' = (-1)' v * even V = V e veil (162) where V cvm is given in equation 161. For ordinary forces III, the potential in odd states is the same as for even. Exchange forces II, on the other hand, have opposite sign in odd states. To determine the behavior of the force sug- TABLF 7 PriOFEitTiES ov a Neutron -Pkotcn System Is o topic Slate Parity Spin s Spin T <TJ ■ CT2 T1-T2 >iS oven 1 -3 1 3 S even 1 +1 -3 !p odd -3 -3 3p oild 1 1 -: 1 +1 gesled by the symmetric theory I, Table 7 of values of crj ■ ffs and ti ■ t 2 has been construe ted for even and odd states of both the singlet and the triplet types. From equation 162 and Table 7, the symmetric theory (I) gives: "f''odd = ~~ /a f'evcii , (M , (163) Todd — ~~ 3 Y even The three types of forces may now be compared with experi- ments by computing neutron-proton scattering at high energy. The energy chosen by Rarita and Sell winger was 15.3 Mev, for which P-wavc scattering begins to be important. The P-wave scattering is to be computed with the aid of equations 162 and 163, SATURATION OF NUCLEAR FORCES 89 which give the potentials acting in the P-state. It should be noted that in contrast to the usual theory in which a single phase shift 5 1 is computed for scattering in the P-state, three phase shifts ij , i} U and 5j 2 must be computed for scattering by the 3 P -, 3 Pi-, and 3 P 2 -states, respectively. The reason for this is that the effective potential well for each of these three states differs because of the presence of the non-central tensor force S&, Tn fact, the operator S 12 has definite values (-4 and 2) for the states ;J P and S P X TABLE 3 Well Depths in the Netjtbal Theory Effective Well Depth with Stale "Ordinary Forces" 3 P 29.2 Mcv (repulsive) 3 P i - 3 5 . 4 Me v (attracti ve ) 3 P 2 - 9.6 Mev (attractive) which occur unmixed and must therefore be eigenf unctions of Si 2- The 3 P 2 -state has a fairly definite value of S v > (—%}, since at 15.3 Mev it is only slightly coupled to the 3 P 2 -state. (See Chapter XIII for a discussion of how 81$ couples states of different L but the same J.) Rarita and Schwinger (Phys. Rev. 59,556, 1941), using equation 161 and the values of S VA just quoted, give the effective well depths for the 3 P-statcs in the neutral theory III as shown in Table 8. TABLE 9 Phase Shifts in 3 Po, 3 Pt, and 3 F 2 States Tlieoi-y 11! VI 12 I 0.074 -0.05-1 -0.017 II 0.531 -0.114 -0.046 III -0.102 0.995 0.073 The potentials of the charged theory II have opposite sign to the tabulated values; those of the symmetric theory I have: opposite sign and a, re one-third as large. (See equations 1(32, 163.) The phase shifts for each of the three theories, using these well depths, are given in Table 9. Note that the phase shifts in Table 9 for theory I are small be- cause potentials arc used which are only one-third as large as for the- 90 QUANTITATIVE THEORY OF NUCLEAR FORCES pries II and III. ( S e e eq uati on 1 G3 . ) N o to al so that th e s igns o f the phase shifts art; opposite in theories II and III because this is also true of their potentials. (See equation 162.) Note further that really large phase shifts occur only for strong attractive potentials, i.e., tjo in theory 11 and 17! in theory III. If the scattering contributions from the 3 P-states are added up with the proper statistical weight (2,1 + 1) the total scattering for 3 JP-states at 15.3 Mev is found to be: 1. II. III. <r(8) - X 2 (0.0038 + 0.0045 cos 3 0) <*{#) - X- (0.103 - 0.002 cos 2 8) <r(8) = X 2 (0,187 + 0.687 cos 2 0) (164) with 4ttX 2 = 0.082 X 10 -34 cm 3 . The scattering is also computed for the (?Bi + ? 'Bi) state. This is added to equation 164, taking proper account of interference terms with the result that the total triplet scattering in barns becomes : I. v(8) = 0.080 (0.983 + 0.002 cos fl + 0,051 cos 2 3) 11. a{0) = 0.746 (0.98G + 0.193 cos 3 + 0.041 cos 2 (?) Ill <f(g) = 1.165 (0.857 + 0.849 cos + 0.429 cos 2 0) (165) The quantities in equation 165 are so normalized that the numbers outside the parentheses represent the total cross sections. A corresponding calculation for the l P and i S scattering gives: 1. a(0) = 0.444 (0.939 - 0.438 cos 3 + 0.182 cos 2 6) IT. &(§) = 0.424 (0.985 - 0.240 cos 8 + 0.044 cos 3 0) 111. a(0) = 0.437 (0.955 + 0.498 cos + 0.1.34 cos 2 8) where the potentials used in the 1 P state were : I. 'V( l P) = -37( 1 .8) = 4-35.7 Mev" II. Fit 1 ?) = -Vi'S) = +11.9 Mev TIL V ( ' P) = V(\S) = -U.9 Mev (166) (107) Note that the difference between a repulsive force (I and II) and an attractive force III is shown by the sign of the teiTn in cos 9 in SATURATION OF NUCLEAR FORCES 01 I 0.621 barn II 0. 6G6 barn III . 983 barn equation 106, which represents interference between the l P and the l S states. The total cross section can be obtained by adding the triplet and the singlet scattering in a 3-to-l ratio. The three theories give hi fractions of a barn the values shown in Table 10. The total cross section should not be used by itself to mate a definite decision between the three theories since it is influenced TABLE 10 Theoretical Xijutr on' -Proton ScATTEHora at 15.3 Mev Theory Total Cross Section Angular Distribution 1 - 0.080 cos + 0.077 coir 8 1 + . 126 cos -f . 042 cos ! 8 1 -h . 032 cos + . 4r>7 cos 2 8 by the range and the shape chosen for the interaction potential. On the other hand, the angular distribution is good evidence for the existence or non-existence of strong P-scattering, and also gives the sign of that scattering— thus providing direct informa- tion about the exchange nature of the neutron-proton force. For comparison with experiment, we may note from Table 10 that at 15.3 Mev, theory I gives a weak backward maximum, theory II a weak forward maximum, and theory III a strong forward maximum. EXPERIMENTS ON NEUTRON-PROTON SCATTERING Total cross sections can bo obtained by measuring the absorption of neutrons in paraffin and correcting for the presence of carbon. Angular distributions have been measured by Amaldi and others (N atu rwissen seh af ten 30, 582, 1942; also Ricerca scientifica 1942), using the recoil protons projected from a paraffin foil. The proton directions are determined by the use of a coincidence- counter "telescope." Proton ranges, hence energies, are deter- mined by the simultaneous use of absorbing foils. In the center- of-m ass system, conservation of momentum re- quires that the neutron and proton leave each other in opposite directions — i.e., at angles 8 and 180° — 8 to the incident neutron, respectively. In the laboratory system, the two particles leave at right angles to each other, and the angle between proton and incident neutron is 90° — 8/2. <Xi QUANTITATIVE THEORY OF NUCLEAR FORCES Amaldi found that the number of protons projected forward was small, corresponding to weak neutron scattering in the back- ward direction, 6 = ISO 3 . This is in agreement with ordinary forces III and in contradiction to exchange and symmetric theories II and I. Amaldi measured R = o-(180°)/V(90 D ), the angles being the neutron scattering in the center-of-mass system. His results are given in Table 11 together with their quoted accuracy. TABLE 11 High-Energy Xeutu m -Proton SciSTBKnJS (Amm.iit) It E (in Mev) 12.5 13.3 14.0 = aCLScryM'JO ) 0.71 ± 0.04 0.53 ±0.03 0.52 ±0.03 The values of 7? at 15.3 Mev computed from the cross-section formulas in Table 10 give for the three theories: I. R - 1.157 II. R = 8.916 III. B = 0.525 (168) On the other hand. Champion and Powell (extension of experi- ments reported in Proe. Hoy. Soe. 183, 64, 1944). using neutrons of similar energy and using photographic techniques, find that the scattering is practically isotropic. However, their experimental data have less good statistics and greater correction factors than Amaldi's.* More definite evidence contradicting Amaldi's results comes from measurements of the proton-proton scattering at energies of 14.5 Mev by It. R. Wilson and collaborators (Phys. Rev. 1947). Al- though these experiments are preliminary, they indicate a slight repulsion in the P-state. They might be reconcilable with ex- change forces or with zero forces in the /'-state, but they appear to fit best to a force of the cr, -0-2 type and they certainly contradict an ordinary force such as would be required by Amaldi's experi- ments. There is, of course, the logical possibility that neutron- proton and proton-proton scattering are different, but. in any ease the present state of this subject is inconclusive and more accurate measurements are urgently needed. * Laughlin .and Kruger (Phys. Rev. 71, 736, 1947) also tind isotropic dis- tribution (at 12-13 Mev).— Note added in pro>>f. SATURATION OF NUCLEAR FORCES 03 If Amaldi's results are correct they imply that the forces in the P-state art; attractive, and they support the theory of ordinary forces III. Unfortunately, this result cannot be easily reconciled with the saturation property of nuclear forces.* * Experiments carried out with, the 184-irich. cyclotron of the University of California at the end of 1946 demonstrate definitely the exchange nature of the forces between neutron and proton. It was shown in these experiments that a neutron of about 100 Mev will produce protons mostly in the forward direction and with energies neatly equal to J 00 Mev. This had been predicted by Wick for high energy collisions between neutrons and protons. If the forces were ordinary forces the proton would in general receive an energy of the order of the depth of the nuclear potential well, i.e., about 10 Mev. On the other hand, if the interaction is of the exchange type, then neutron and proton will change roles: the neutron will re Lain an energy of the order of 10 Mev and the proton will take almost the entire energy. When this note was written it had not been established whether the forces are of the pure exchange type or of the type corresponding to the symmetrical meson theory.— Note added in proof. XV. SKETCH OF THE MESON THEORY OF NUCLEAR FORCES This theory is presented although it Iras so far not given any results in quantitative agreement with empirical facts on nuclear forces. However, it may give a valuable point of view. The Coulomb force between two charged particles can Ik; ex- plained in terms of the interaction of these particles with the elec- tromagnetic field. Similarly, the force acting between two nucleons might fee described by a meson held surrounding the first particle which acts on the second. Moving charges produce a radiation field which can be quantized and described in terms of photons. The "quanta'' surrounding a nuclear particle arc called mesons. Yukawa, in initiating the meson theory (Proe. Phy si co-Math. Soc. Japan 17, 48, 1935), suggested that if the mesons are given a finite rest mass m. the range of forces arising from the meson field will be h/mc, the Compton wave length for the meson. If the range of nuclear forces is assumed to be 2,8 X 10 _1B em, the meson rest mass should be about 140 electron masses. Particles with about this rest mass were discovered in cosmic rays two years later. In the meantime, Erode- and Fretter have determined the rest mass to be 202 ± 10 electron masses, giving a range of 2 X 10' I,! cm. To determine the nature of the meson field and the correspond- ing nuclear forces, an equation analogous to V- $ = — 4irp must be written for the static part of the electromagnetic field. A rela- tivist! c equation suited for particles with no spin and a finite rest; mass m is the Klein- Gordon equation : ?V + (l/nV)[(E - TO 2 - (mc 2 f]4t = -Ittp (169) with E = ih(d/di) (169a) where p in this case is proportional to the density of nucleons. In free space, V = 0. For a static meson field, according to equation 169a, we must put E = 0. Furthermore;, if there is one point- nucleon at the origin, the Klein-Gordon equation, becomes W% - {mefMH. = 4ir E / 1 5(r) (170) 94 MESON THEORY OF NUCLEAR FORCES 95 where o represents the Dirae ^function , and g L is a constant replacing the electronic charge in electrodynamics. The solution of this equation is $ = -(d/r) exp [-(jn.c/h)r\ (171) and the potential acting on a second nucleon is given by: V = tyfr 0"2) where ffi and g 2 are the effective nucleonic "charges" or coupling constants. The Yukawa scalar meson theory just described produces the required range for nuclear forces. Since in this theory the nuclear particle does not change its nature (i.e., charge) we find that according to tire theory the neutron-neutron, neutron -pro ton and proton-proton forces are all equal. However, the theory does not explain the spin-dependence of nuclear forces. Furthermore, the forces are all "ordinary," whereas exchange; forces were found to be necessary to explain the saturation of nuclear forces. Since tire mesons discovered in cosmic rays were; all charged either positively or negatively, a theory of charged mesons was developed. According to this theory, the following reactions can take place: p^# + A + m N ^ p + ^- (]73) Thus protons and neutrons can transform into each other by the emission or absorption of positive or negative mesons. The interaction between two particles, 1 and 2. can take place, for instauee, by the following scheme: Pi -* Ai + ^ -Y 2 + M + - > i\ (174) It is clear that such an interaction can only occur between a proton and a neutron, not between two like particles. This is in contradiction to experimental evidence and rules out the charged meson theory, at least in tin; case of weak coupling between nucleons and meson field (small value of g). Further, the charges of particles J and 2 are exchanged in the process of emission and reabsorption of the meson: therefore; this meson theory leads to a force of the charge exchange; or Heisenberg type. This, while giv- ing saturation, is in contradiction with experiment (Chapter XIV). To explain the neutron-neutron and proton-proton forces which are missing in the charged theory, a symmetric scalar meson theory was developed, containing neutral, positive and negative mesons described by three functions #i, fa, and ifo. To get spin-dependent 90 QUANTITATIVE THEORY OF NUCLEAR FORCES V = g i + °V°"2 (175) nuclear forces, the meson field must further depend on the spin of the nucleon which generates the field. This is achieved by intro- ducing into the Tlamiltonian of nucleon plus meson field, an inter- action energy containing the factor cr ■ grad $ where u is the nucleon spin. In this case $ must be a "pseudoscalar" since <r is an axial and grad a polar vector. (A pseudosealar changes sign -when the sign of the time is reversed, or on inversion of the Spatial coordinates; under Lorentz transformations, it is invariant.) Solution of the symmetric pseudosealar meson field equation led to an interaction energy between two nucleons of the form i r /3 :v A -Tl-T 3 s ia i-.|-.-+-i 3 _ \r r r / where i± = mr;/h. The term in o-j - cr 2 provides the spin dependence of nuclear forces, and the tensor force S l2 explains the existence and sign of the quadrupole moment. All these features are in qualitative agreement with experiment, as shown in the preeeding chapters. Unfortunately, the high singularity of V at r = makes it impos- sible to solve; the Schrodinger equation. Two ways of saving the situation have been suggested: (1) to cut off the interaction at some finite radius r u , i.e., to give the neutrons and the protons a finite size, or (2) to mix two meson theories in such a, way as to eliminate the undesirable singularity. Tie assumption of finite sources (1) unfortunately caimot be formulated in a relativistic invariant way. Furthermore, use of the rigorous relativistic interaction between nucleon and meson field leads to the reappearance of terms in \/r and l/r s in the "mixed" theories, in higher approximations. Therefore there are at present no trustworthy results of the meson theory of nuclear forces. It should be noted that many of the statements made about the spin and charge dependence of the nuclear forges have to be modi- fied if the coupling between nucleon and meson field is strong, i.e., if many mesons are emitted simultaneously. The coupling con- stant for an electromagnetic field is (?/lic = MV 7j "■ SB6&EI value, whereas that for the meson field cf/hc ca 14 or % is considerably larger. The divergence of the interaction at sma.ll distances makes the interaction effectively even stronger. For this reason, much effort has been spent to treat the strong coupling problem in meson theory, but so far no results have been obtained which throw light on the problem of nuclear forces. C TOPICS NOT RELATED TO NUCLEAR FORCES XVI. BETA DISINTEGRATION In Chapter VI, experimental evidence was given for the hy- pothesis of the production of neutrinos of rest mass and spin }/ 2 in 0-decay processes. This assumption made possible the conserva- tion of energv and spin. The first detailed theory of the process was given by Fermi (Zeitschrift fur Fhysik 88, 161, 1934). A modification which seemed necessary but was later abandoned was the work of Konopinski and Uhlenbeck (Pbys. Rev. 48, 7, 1935). A summary is given by .Konopinski (Rev. Modern Phys. 15,209, 194.3). Fermi introduced a new interaction between the nucleon and the two light particles, electron and neutrino. His interaction was chosen in analogy with the interaction between charges and electromagnetic field in quantum electrodynamics. (This analogy was also used in the last chapter in connection with the meson theory of nuclear forces.) The heavy particles are to act as sources and sinks of the light particles. If the Hamil Ionian of the interaction between the proton, neutron, and electron -neutrino fields is H ; then the number of transition processes per unit time is (2^/h}|J> fitI .*^,,^ 3 • pQB) (176) where p{S). — the number of final states of the system per unit energy interval =s initial state of the system = u^. = initial state of the nucleon. = Mfin. ' &fee. ' <?n. = Spal state of the system = (final state of nucleon) ■ (final state of electron) ■ (final state of neutrino) . to Fermi's assumption for // was essentially Jfe* Hto„ (It - gfu m * fee* <^.ifin. & (177) (neglecting relativistic corrections which are important only if the heavy particle has high velocity) where fee. arul S8t. arc to bt; 97 98 TOPICS NOT RELATED TO NUCLEAR FORCES evaluated at the position of the nucleoli, and therefore the integral is over the coordinates of the nucleoli alone. This is similar to the case of electrons and light: a charge can only interact with a light quantum, when they are at the same place. The constant § which determines the strength of the interaction must be found from experiment. It has the dimensions erg ■ cm 3 , since 1^1^. and <p n . are Lo he normalized per unit volume. Note that we use ^i^,*, bat p a , (without a star). This corre- sponds to the emission of an election but the absorption of a neutrino. However, this absorbed neutrino can be taken from a state of negative energy which corresponds to the emission of an '■'antineutrino." Owing to the absence of charge and magnetic moment, an antineutrino is equivalent to a neutrino. The formu- lation (177) is therefore equivalent to the emission of an electron and a neutrino, and it is a mathematical convenience to have formally one particle absorbed and one created. The positron emission would be described by #ejes.*k* Since the neutrino has very little interaction with anything, its wave function may be taken as a piano wave. If p n . is the; mo- mentum of the emitted antinouirino, then — p n . is that of the absorbed neutrino of negative energy, and Pa. = T ! -exp (-i p ru - r/fi) (178) where V is the volume of a box in which the wave function is normalized. The factor V~ 1 ' 2 may be omitted if a unit volume is used for the normalization. $&«. should be a Coulomb wave func- tion; but if Z the charge number is small, the Coulomb energy of the electron can be neglected in comparison with its kinetic energy and. a plane wave can be used for the electron wave function. The number of final states per unit energy is P (m = (Volume element of mo- (Volume element of mo- mentum space of electron) men turn space of neutrino) (Volume of phase space per (Volume of phase space per electron energy state) " neutrino energy state) XdE = (Jeta 2 #ete $»ri«J IP". 3 # B . doi n )/(2rhf d.E n , where do> elvi .da) n _ are elements of Solid angle. (179) BETA DISINTEGRATION The result for the transition probability of an electron into dS d[iC . and solid single i&f&w (integration over all directions of the neutrino has been carried out) is G 2 mc? I r §^~h~P^ exp -i(pn.+P C ]o .)- 2 , m rfre( e 3 -l)^ U -0 2 rfe- (180) with G = (f//mc 2 )(h/mc) % e = E^Jmtr, V r - I. = p^/jrac, eg = E^, llihM Jmc 2 . A plane wave has been substituted for the electron wave function. Just as in the theory of atomic transitions, there will be selection rules for /3-deeay processes. If p„ h: ... and p n _ are both of the order of magnitude mc. as is usually the case, the exponent (p„ -f- p (!let .,) ■ r/h will be of the order of magnitude; 4 X 10"" cm h/mc 3.85 X I0~ n cm 1 100 (181) (R = nuclear radius; medium-weight nuclei have been chosen.) Thus, exp [i(p n . + p ek .c.) ■ r/h] will be nearly 1, and the matrix element in equation 180 reduces to M = J&fia.**^. dr ; i.e., to an expression depending only on the state of the nucleon before and after the transition. M is determined by the nuclear wave func- tions. In parti cular, the orthogonality of the nuclear wave func- tions for states of different angular momentum I gives the selection ni,e: M fi implies AT = (182) Such transitions are called allowed. Transitions for which M = are called forbidden; in this case the exponential in equation 180 must be expanded in a power series; the order of the forbidden transition is the number of the first term in this power series which gives a n on -vanishing result for the matrix element. Because of the estimate (181), the probabilities should decrease by a factor of about 10 4 with each order. ALLOWED TRANSITIONS The only depend ence of the allowed transition probability on the electron energy is through the volume clement in momentum space. The energy spectrum of electrons is therefore KB de ~ el/e s - Ifo - tf d* (183) 100 TOPICS NOT BELATED TO NUCLEAR FORCES Since eo is unknown, the experiments have to yield a value of t#, while giving a check on the theoretical spectrum.. This is easily done by making a "Kurie plot." In this plot, the quantity F(e) = V;V(f)/£(i 2 - 1) H (184) According to hould vie Id a F<t) (as observed) is plotted against the energy & equation 183, F(e) <--■ e () — e; therefore the plot straight line which cuts the e-axis at eg. The only nucleus which checks this proportionality exactly is In 114 , measured by Lawson and Cork (Phys. Rev. 57, 982, 19-10). Hero to = 1.99 M.ev (which is high enough to make the ex- periments on the /3-rays fairly easy) and the lifetime is 72 seconds. Luckily this short- lifetime /3-dccay follows a 50- day-lifetime 7-decay (isomeric transition; see Chapter IV). There are experimental dif- ficulties in flic measurement of the energy spec tin of most Thick target 0.615 Mev E Fig. 13. Kurie plot of the positron spectrum from Cu . other /^radioactive nuclei which result from either the low energy of the electrons or the short lifetimes. Cu M measured by A. W. Tyler (Phys. Rev. 56, 125, 1.939) emits both positrons and elec- trons. The positron spectrum was measured both for thick target and thin target (thick and thin relative to the electron range). The Kurie plots are shown in Pig. 13. It is not known whether the portion AB of the thin target curve is spurious or results from another decay process (to an excited state of JMi* 54 ) with a very low energy limit. The thick target curve is typical of the experimental evidence which lead Konopinski and L'hlenbeck to introduce then alterna- tive theory (Phys. Rev. 48, 7, .1935). They proposed using the time derivative of the neutrino wave function &&J&i in the transi- tion probability instead of ip. Since &/>/ dl o-< (sq — e)^ this led to spectrum eVe 2 - l(«u - <0 4 de A 7 k-u de>~^ e - * (185) thereby moving the maximum of the spectrum to lower electron BETA DISINTEGRATION 101 energies. To make a Kurie plot of this, the fourth root must be used in equation 1S--1 instead of the second. Many of the experi- mental data on thick targets then give straight lines but very high values of €<>. Later experiments us ing thin targets showed that the Kurie plots according to the Konopinski-TJhlenbeck theory dropped off, as shown in Pig. 11, which demonstrated that the straiglrt-line portion was accidental. Also, when the mass differ- ences of nuclei became better known, the values of <e given by the Konopinski-Uhlenbeck theory were shown to be much too Mgh in all cases but that those given by the Fermi theory agreed with the measured mass difference. A' 13 measured by Kikuchi et al. (Proc. Physi co-Math. Soc. Japan, 21, 52, 1939); Lyman (Phys. Rev. 55, 1123, 1939); and VV--V Fir.. 11. Typical Kurie plot of the Konopinski- Ohlenbcck theory. Townsend (Proc. Roy. Soc. A177, 357, 1941), is one case in which the use of very thin targets still did not gore a Fermi distribution. To account for such spectra it is usually assumed that several decay processes an; taking place simultaneously, leading to various energy levels of the residual nucleus. With N 13 — » C li + § this is confirmed by the observation of a y-ra,y of about 280 kev by Richardson (Phys. Rev. 55, 609, 1939). This y-ray is attributed to the transition of the residual nucleus C 13 from its excited to the ground state. Unfortunately, various experimenters disagree on the relative intensities of the 7-rays and of the two compo- nents of the /3-spectrum, and on the value of the upper limit of its lower-energy component. Coulomb Field. In expression 183 for the electron energy spectrum no account has been taken of the Coulomb field. The correct spectrum has a greater electron density at lo w ener gies. There is no zero for e = 1 because the factor V« -l~s (velocity) in the density of states Is canceled by a 1/v in the charge ■ I 102 TOPICS NOT BELATED TO XL" CLEAR FORCES density of electrons at the nu ele us. The result i ng electron spectrum is showa m Fig. 15. Far positrons, fewer of low energy should be expected than the number given by expression 183 because of the repulsion of the positrons in the Coulomb field: The Coulomb wave function of the electron in expression 177 has a facto c cxp ( — 2-irZcr ,/hy) , which lowers the transition probability considerably for low velocities. There are some disturbing measurements by Backus (Rhys. .Rev. 68, 59, 1915) on the ratio ., in the Cu fil ^-transitions: Fig. 15. Energy distribution of jS-mys with Coulomb field. of positrons to electrons, N+/N. Cu 64 - f \m m + ,3- [Zn^ + f a calculable way at N+/N"_ should be smallest and behave in low energies; the experimental values were compared with the theoretical prediction but the value of N_j_/N_ was found to be ten times greater than predicted. These measurements should be repeated. The disagreement can hardly be attributed to a failure of 0-ray theory because the ratio of positron emission to /C-electron capture was found to be in exact agreement with theory (Scherrer et ah, Phys. Rev. 68, 57, 1.9:1:5}, and this ratio Involves parts of the theory very similar to those in Backus' experiment, LIFETIMES IN ALLOWED TRANSITIONS The total transition probability, or reciprocal of the lifetime, for /3-ray emission is found by integrating over the energy distribu- tion (equation 180) to Ire 1/V b (G 2 /2tv*) • (mcVh)|ilir/<-(^) (186) G is a dimonsionless constant describing the strength of the inter- action between electron-neutrino and the heavy particles: M is the matrix element for the transition : M = fu nn *(T)u in _(r)e- i( ^+*< ] -* /h dT (187) ! BETA DISINTEGRATION F(e ) is the integral of the distribution in energy f(& = £*W^ - *& - *f^ 103 (188) where e is the total energy available for neutrino and electron, including rest mass, in units of the electron rest energy. W (*$) varies rapidly with e (1 , being approximately equal to ( ! /SO) e s for e » 1 aurl to 0.216(fF — l) y ' for sg nearly unity. Thus r de- creases rapidly with increasing e , but not as fast as in the case of a-decay, where the transition probability is proportional to an exponential of the energy. In Chapter II it was pointed out that in natural a-deeay a factor of 2 in energy is equivalent to a factor of 10 -20 in lifetime. The matrix element M is in general not known because we have very scant knowledge of nuclea.r wave functions. Even if we know that the transition is allowed, we can in general say only that \M\ is between zero and one. However, in some cases the value of M can be guessed to some- what better than order of magnitude. For allowed transitions (A/ = 0) , we have M «/%*.%, 4r (189) M will be near unity when the wave functions u fill . and u- m , are nearly alike. Such is the ease for ,6*-transition between mirror nuclei (Chapter II) (for which also the selection rule Al — is likely to be fulfilled). Three examples of allowed transitions in mirror nuclei are given in Table 12. The product iFfa) is remark - TABLE 12 Allowed Transitions in Mihhob Nuclei Reaction t = half-life eo tFUn) H 3 -a He 3 + (?~ + " 10* m 1.03 1400 C u -* B" +fT + v 1200 sec 2.S6 3500 So* 1 -» Ca 4! + D+ + v . <J sec 10. OS 2500 Source: Konopinski, Rev. Modern Phys. 15, 209. ably constant, confirming the theory underlying equation 186. This constancy exists in spite of t varying by a. factor as large as 10 s . Furthermore, it, is reasonable that, tF is somewhat smaller for the first situation than for the other two, for in a nucleus 104 TOPICS NOT RELATED TO NUCLEAR FORCES containing only three particles we would expect u fm , and i%, to be more nearly alike than in the heavier nuclei; so that | M \ would be closer to unity in the light nucleus. It is interesting to note that the Konopinski-TJhlenbeek theory of /3-decay predicts variation by a factor of JO 5 between the prod- ucts i-F for the various reactions in Table 12. For nuclei of intermediate mass, the Coulomb repulsion already introduces considerable asymmetry between the numbers of pro- TABLE 13 Allowed Transitions in Intilkmediately Heavy Nuclei Reaction g?5 _> C1 3S + ?- + v Cu fjl -> Zn M + $~ + v Cu M -> Ni i4 -T- 3+ + v In 117 -h» Sn 1 *' 4 + ' 19,000 66,000 22,000 140,000 tons and neutrons (there are no more mirror nuclei), and pre- sumably even greater' differences, between neutron and proton wave functions in the nucleus. Thus, even for allowed transitions, smaller matrix elements are expected for intermediately heavy nuclei than for light, mirror nuclei. T his is borne put by the data in Table 13. In the heavy, naturally radioactive nuclei the matrix elements TABLE 14 Allowed TsajtskSOKS in Naturally Radioactive Ndclf.i Emitter tF(ta) RaB 50,000 UX S 270,000 are in general still smaller. This is borne out by the data in Table 14. Assuming \m\ ~ 1 for the lightest., mirror nuclei, can be cal- culated from Ft. The result is G « 10" (190) This corresponds to g ~ 10 ' ls erg • cni a . The smallness of this coupling between electron-neutrino and the heavy particle is BETA DISINTEGRATION 105 what makes (3-decay take place so slowly compared to other nuclear reactions, except some a -radiation. It is safe to say that ji-rayti are not emitted during nuclear collision*, but only at com- paratively long times afterwards. For example, the lifetime of protons in the sun due to the reaction II + H -» D + r + v (191) is about 10 11 years, even with a density of about 100 and a tem- perature of 2 X .10 T degrees C. (See Bethe and Critchfield, Phys. Rev. 54, 248.) Even so, this reaction presents about the best opportunity for /3-decay during a collision. The long lifetime of the proton in the sun indicates an extremely low probability of /'i-decay per collision. The most fundamental (3-decay is that of the neutron n —> II + j3 + v (192) The matrix element for this reaction should be exactly unity, as the wave function for a single proton ought to be the same as that of a single neutron. Measuring the lifetime of this reaction should give an exa.ct value of G. However, this reaction is hard to observe as the neutrons are removed much more rapidly by other means (capture, diffusion) than by the above reaction. Using the value of (J found above, the half-life for the reaction (192) should be about 15 minutes. There is hope of making the measurement with the large neutron fluxes now available in piles. LIFETIMES IN FORBIDDEN TRANSITIONS The second term in the Taylor expansion of the exponential in the matrix element (1ST) will give a non-vanishing integral when A/" = zhT. which transition was forbidden in the first approxima- tion. Similarly, A/ = ±2 transitions become possible with the third term in the expansion, and so on. For t = 2, the argument of the exponential averages about 1/100 over the range of the heavy par tick; wave function, so that \M{AI = =L.L)'| 2 might be expected to be about JO" 1 times \M(&I = Q)| 3 . Actually, the true wave function for an electron in the Coulomb field varies faster than the plain; wave approximation used in equation 187, and the factor 10" 1 becomes about 10~" for medium and heavy nuclei. This correction does not help the higher forbidden transitions SO 106 TOPICS NOT RELATED TO NUCLEAR FORCES much as the first. Higher ^ makes all forbidden transitions more probable. Table 15 quotes experimental data from Konopinski for forbidden transitions in light nuclei. TABLE 15 IIaIjF-Ltves in FoRBIBOEN Transitions Emitter t = Half-life m First Forbidden, Transitions &fa) Li s 0.9 sco 24.5 2.S X 10 5 Ne 23 40 see 8 10 r ' Second Forbidden Transitions pn 1.2 X 10 B see 4.37 Higher Forbidden Transitions 8.6 X I0 7 B<; m KV^SCri 2.1 10 H K « 5.10 16 sec 2.4 10 11 Source': Konopinski, Rev. Modem Phys. 15, 200. GAMOW-TELLER SELECTION RULES Tli ere is good evidence that the selection rule AT — for allowed transitions is not generally adhered to. One example is the K- eapture reaction ^ + K ^ L . 7 + p (m) Li 7 is produced both in its ground state and in an excited state about 440 kev above the ground state. The experimental ratio of number of transitions to the ground state to number of transitions to the excited state is about 1.0 to .1. This is about equal to the calculated ratio, using equation 186 and assuming \M\ equal for the two cases. From this and the absolute lifetime it may be concluded that both transitions are allowed. However, we do not expect both states of Li 7 to have the same value for I. The best assumption is that the two states form a /'-doublet, with I = Yi and I = % for excited and ground states, respectively. Thus AT can certainly not be zero for both transitions. Another example is the reaction: He* -» Li 6 + /T + v (194) Li a can be thought of as an ct-particle plus a deuteron. The .a-particle has / = 0, and the deuteron has I =» 1. We expect, BETA DISINTEGRATION HIT therefore, that Li a has I = 1, in agreement with experiment. In the same picture. Tie 6 is an a-partiele plus two neutrons. In the "ground state," the double neutron should have spin zero (ef. Chapter XII), so that the same argument gives 7 = for He*. An additional argument for this is that all nuclei containing even numbers of neutrons and protons have zero spin as far as they have been investigated. Thus AI = I, and the transition is forbidden. But the experimental lifetime of the reaction shows that it is "allowed," There are similar situations in the /3-decay of C i0 , F 18 , and Na 3U . So it seems that there can be allowed transitions with A 7" = 1. Gamow and Teller first allowed how this can come about. They said that in considering possible interactions, one ought to include all relativistically invariant combinations of the four wave func- tions, u- m ., tifijj., &jfeM and 6,... For two wave functions, let us say \f/ and 4>, there are five combinations which are co variant under Lorentz transformations : 1. Scalar: ^* j3 <i> (Fermi theory). 2. Polar four vector, with components: ^* 6, f* a </>. 3. Tensor: $*■&&■$, V P a <t>- 4. Axial vector: $* cr <*>, #* 7s *. 5. Pseudoscalar : $* ,8 y 5 <j>. where fi, a, and y 5 are Dirac operators and a is the usual spin operator. (For details, see Konopinski's article.) To obtain a relativisficaily invariant interaction, the corresponding combina- tions of the wave functions of the light and of the heavy particles must be multiplied; for example, the tensor combination of the light particle wave functions with the tensor combination of the wave functions u m , and i( fin . of the heavy particles. In this case the Hainiltonian becomes: V (tensor) = (f*j3<r4) ■ (u&^&TMia) (195) (The transition is still treated as though an antineutrino is emitted.) Since the heavy particles are non-relativistic, the Dirac operator ,3 for them is equivalent to unity; therefore, the net effect of equation 195 is to place the operator cr between the heavy particle wave functions fe. and « fln- . Therefore, the matrix element for allowed transitions is now J ■'u-fi 11 .*<nii il , dr, and this may be different from zero if the total sphi I changes by one unit, mmm 10S TOPICS NOT RELATED TO NUCLEAR FORCES or by zero, in the transition. Thus M = ±1,0 can be "allowed" for the tensor interaction. The axial vector interaetion gives the same selection rule as the tensor, Al = 0, ±1 (106) From the experimental data it seems that these Gamow-Teller selection rules are correct. For instance, they explain the results for He 6 , C 10 , F 18 , and Na~. However, the reaction Be in -» B 10 + &~ + v (197) differs from (194) only by the addition of an at- particle, so that &I = 1 may again be expected for this reaction. But experiment shows that this is forbidden. The same is true for the reaction C 1 n m + £-- + v (198) which differs from (191) by two a-partieles. Thus the Gamow- Teller selection rides, while;, explaining more than the Fermi rules, still are in contradiction with many of fetes data. K-capiure. The theory for A'-capture lias been worked out, and is in good agreement with experiment. Scherrer el al. (Phys. Rev. 68, 57) have measured the ratio of Z -capture processes to positron-emission processes for Cd 107 (or loy ?), with the result: 320 =fc 20. The Fermi theory predicts 340. (The Konopinski- Uhlenbeck theory gives 20,000, and is conclusively ruled out.) XVII. THE COMPOUND NUCLEUS In tins chapter, we are no longer concerned with the determina- tion of fundamental nuclear forces, but with the more practical problem of predicting cross sections for nuclear reactions, par- ticularly those involving heavier nuclei the quantum states of which are not known precisely. On the other hand, the presence of many nuclear particles will make statistical methods practical, and these are used in the theory of the compound nucleus. The concept of the compound nucleus was initiated by Bohr m 1935. In order to get a clear picture of this concept, we shall examine the difference between nuclear collisions and atomic collisions. For collisions between an. atom and a particle of high or moderate energy, the Born approximation is valid because the incident particle passes right through the atom practically undisturbed. Slight deflections, inelastic collisions, and emission of radiation are progressively less likely processes. The reason that particles are likely to pass right through is that the atom is a loosely bound structure. Another way of saying this is that the interaction of atomic electrons with, say an incident electron of several thousand volts, is much smaller than the incident energy- -which is precisely the condition for validity of Bom's approximation. Nuclear interactions, on the other hand, are of the order of 20 Mev, which is much greater than the kinetic energy of the incident particle normally used, i.e., several Mev or less. This is precisely the opposite of the conditions required for Born's approximation. Here, the interaction energy is more important than the kinetic energy. Another difference: An electron striking an atom can be re- garded as interacting with the average "Hartree" field of the atom. This approximation is valid because the interaction with a single electron is much smaller than the average interaction with all the electrons. On the; other hand, the short range and the saturation character of nuclear forces require that, nucleons interact only with a small number of neighbors. Tims individual interactions 109 11.0 TOPICS NOT .RELATED TO "NUCLEAR FORCES will be of the same order of importance as the average total inter- action — and it will not be permissible to replace tire nucleus by an average field. The Bohr picture takes advantage of these large interactions and describes them in terms of a compound nucleus. Tire theory makes the following statements; 1. Any 'particle which kits the nucleus is caught. A new nucleus is formed called the compound nucleus. The reason for this is that an incident particle will interact with one or two nucleons, transferring much of its energy to them and thus to the nucleus, before penetrating it appreciably. Then it may no longer have sufficient, kinetic energy to escape the attractive nuclear forces, and is therefore caught. 2. The compound nucleus is long-lived compared to the natural nuclear time. (This is the time for a neutron to cross the nucleus — ■ cm say 10 ~ r2 cm/ 10" -' - =* 10 ~ 31 second.) The reason for this is sec that the compound nucleus, which is in air excited state (excitation energy above the ground state = incident energy T binding energy of oire particle), will live until this excitation energy, or a reasonable fraction of it, is concentrated again on one particle. 3. The final break-up of the nucleus is independent of the mode of formation, i.e., regardless of how- the nucleus was formed there will be definite probabilities for decay into each of several possible residual nuclei. This can be explained in terms of the long life- time of the compound nucleus during which, complete statistical equilibrium, is assumed to be established — thus the nucleus forgets how it was formed; formation and disintegration can be regarded as independent events. For example, the ordinary Al nucleus ( l3 Al 2 ') can be formed as a "compound nucleus" in a highly excited state from any of the (199) reactions ; u Na 23 + W -» 1S A1 37 excited" 12 Mg 2E + Tl 2 -> 13 A1 27 excited 12 Xlg 2fi + H 1 -» "AF excited "Al 27 + t -> IS AF excited. The compound nucleus can then decay back, reversing the reac- tion, into any of the nuclei just mentioned, or also into AF + %, THE COMPOUND NUCLEUS 1.11 with a definite probability for each which is the same for all modes of formation. The residual nuclei may also be left in excited states, with probabilities which are also independent of the manner of formation. Formation of Compound Nucleus. The cross section for forma- tion of the compound nucleus cy may be written in the form H = tvR.% (200) where H, is the nuclear radius, and £ is a useful parameter, called the sticking probability, which is defined by this equation. For fast nuclear particles, i.e., X «R{\ ~ 10" 12 cm for 200-kv neutrons), the classical geometrical approach is valid since the uncertainty in position of the particle is only X. The cross section for capture of fast nuclear particles is certainly not greater than vB 2 since the interaction is negligible ii the particle passes at a distance from the nucleus. For slow neutrons, however, cross sections greater than irR 2 are possible since the position of the particle is poorly defined. To get a sticking probability which is always <1, the definition is revised. We define the contribution ffL to" the cross section due to particles of orbital momentum I, andset *, - (21 + 1)*X% (201) Then from general principles of quantum mechanics, £j must be less than (or equal to) 1. Moreover, equation 201 reduces to equation 200 for high energy since all values of I up to R/X will contribute appreciably (cf ." Chapter IX, p. 38) ; £ is a weighted average of % Neutrons were used in the above discussion to avoid questions involving penetration of the potential barrier winch would arise for protons and a- particles. The Bohr statement, that, any particle which hits the nucleus is eatight, is given more precisely by the equation In other words, the sticking probability approaches 1 at, high energies. This statement has been checked experimen tally with high -energy neutrons especially by Amaldi and co- workers, by Sherr, and by Graham and Seaborg. They find cross sections of about it/ & irft 2 , with E given, by a formula similar to equation 3, in good agreement with other methods of determining nuclear radii (sec Chapter II). 112 TOPICS NOT RELATED TO NUCLEAR FORCES Disintegration of Compound Nucleus. The probability that the compound nucleus will disintegrate in a particular way is related to the cross section for the corresponding inverse capture process with some factors containing Ihe density of initial and final states. This follows from considering a statistical equilibrium condition between the compound nucleus and all the possible states of all the residual nuclei into which it can disintegrate (similar to Chapter XI, p. 60). In. equilibrium, the number of nuclei present in a small energy range between E and E + dli will be proportional to the density of states p(E) in that energy range, and to a Boltz- mann factor. Since energy is conserved in the total system, the Boltzniann factors cancel out and the condition for equilibrium takes the form PjtWx-fji « m&B^jk (203) where p A and p Ti are the densities of initial and final states of the system at corresponding energies, and the W's represent prob- abilities for the direct and inverse processes. For our process, A is the excited compound nucleus 'rath a density of states pa(Ea) = I/Aaj where D is the average separa- tion between neighboring states, at an energy E A above the ground state of A- (Each state is counted according to its statistical weight.) Wa-*b is &hs probability of disintegration of the com- pound nucleus into a definite state of the residual nucleus B with energy E$ above its ground state, with the emission of a particle (say neutron') of energy E. We^a is the probability that nucleus B will capture this particle of energy E and produce a compound state of excitation E A . Finally, ph(Ejj) gives the number of states between E and E + dE available for tin: outgoing particle, viz. 47T-/J 2 PS v(2*Ky s (203a) with p and v the momentum and velocity of the outgoing particle. We now use the relation between the capture probability and the capture cross section, which is Wg^j vc f (E) (204) for one neutron in a. box of unit volume moving with velocity v => (2Efm) ' /2 , and the relation between the excitation energies E A and Eg, Ea E - B (205) THE COMPOUND NUCLEUS 113 where E is the energy of the outgoing particle and B its binding energy in the unexcited nucleus A. Using all the relations just given, and setting I = in equation 201 (other I give veiy similar results), we now have a relation by nieans of which the disintegration probability Wa~*8 — iVh can be computed in terms of the sticking probability £g for the inverse capture reaction : n//;bi)(TVh) = m *$fe ( 20e ) or, inserting 203« and simplifying: T b /Da = fr/2* ( 206a ) This important equation relates the disintegration probability Fb, leading to a definite state of the residual nucleus, to the level spacing D A . For high energies, 6s approaches 1; for low energies it is proportional to the velocity v of the emitted particle. Both Da and T B can be deduced from experiment; D A and % B can also be estimated from various statistical models for heavy nuclei (Nuclear Physics B; Weisskopf, Phys. llcv. 52, 295, 1937; 57, ■172, 1940). The disintegration probabilities tjj/fi >"'e also related to the widths of the resonances observed in these reactions: since; the total decay probability is V/h (i/h)^ B the time dependence of the wave function is of the form -iEl/h -lt/2h _ (J -i{li!- f]'lf/h (207) (208) (Note that the absolute square of the wave function gives the occupation of the state and decays according to equation 207.) Equation 208 has a Fourier transform * the absolute square of which is: (209) (E' ~E) 2 + (r/2) £ Thus T has the same dimensions as E and gives the width at half- maximum of the level, or resonance line. The quantity T B repre- * Taking the Fourier t ns.n?f mid with respect to time of a- time-dependent wave function gives the wave function 4i{B') in energy space. __ 114 TOPICS NOT RELATED TO NUCLEAR FORCES sents a partial level width, i.e.. the contribution to V arising from the disintegration into a definite end state B. Since the compound nucleus must eventually decay, the cross section for a reaction ending in state B is given by the cross section for forming the compound nucleus, times Fg/F. Thus (210) and for fast particles : ays = °7 iwr ujb = rS'| Tg/T (211) CONCLUSIONS ABOUT NUCLEAR REACTIONS Energy Distribution of Emitted Particles. From equation 206a we see that V B is almost the same for any final state B, since the sticking probability k B is a slowly varying function of the energy of the outgoing particle. This information is useful in predicting the energy distribution of the emitted particles. For example, if we consider the inelastic scattering of neutrons Z A + n->Z A -'^Z A ^n (212) and make use of the fact that the density of states in the residual nucleus increases rapidly with excitation energy, then we see that the residual nucleus will most likely be left in a fairly high excited state and the emitted neutron, will come out with low energies. The fact that emitted neutrons come out with greatly reduced energies has been experimentally confirmed for many target nuclei. Lead forms a notable exception to this rule. The reason for this may be that the first excited state in this instance is quite high — so that this rule would not be confirmed unless higher energy incident neutrons are used. In fact, the incident energy must be high enough so that the residual nucleus B possesses a great many levels with an excitation energy less than the incident kinetic energy E , in order that the statist! eal considerations used may be valid. Shadow Scattering. In neutron-scattering experiments a purely wave-optical effect must be considered at high incident energies (X « B). for which we have said the capture cross section is x.8 . In this case, the nucleus can be regarded as a black sphere of radius R which casts a shadow. This is described in the language of wave optics by saying that just enough light is scattered in the THE COMPOUND NUCLEUS 1 15 forward direction to cancel the incident beam. This would mean a cross section for shadow scattering of irR 2 . Fur therm ore, to cancel the incident beam behind the sphere, this shadow scattering must be of the same energy, i.e., it represents elastic scattering. According to an elementary wave-optical argument, the shadow Scattering will be mostly confined to an angle X/fi from the forward direction. In the case of light, for which normally X « B, the shadow scattering is not easily measurable since the shadow extends prac- tically to infinity. In the nuclear case %/B is, say, y 6 or %, so that the umbra or region of complete shadow extends only a short distance back of the nucleus, certainly not as far back as the measuring apparatus. Thus il is possible to make measurements outside the main beam but still at small enough angles to it to obtain the elastic shadow scattering. The existence and general features of shadow scattering have been confirmed experimentally by Kikuehi ot ah, Amaldi et ah, and Backer. Charged Particles. The emission of charged particles such as protons requires the penetration of a potential barrier. This penetration probability is similar to that given in the theory of a-decny and is quite small unless the emitted protons have energy nearly equal to, or greater than, the barrier height B. Thus, in a rough way, we may say that the protons must leave with a mini- mum energy B. This would leave the residual nucleus at a lower energy than if neutrons were emitted. Since the density of residual nucleus states decreases rapidly with decreasing energy, the: probability for proton emission will be much smaller than that for neutron emission because of the fewer number of states available, especially if the nuclear charge is high and the available energy low. y-ray$. The emission of 7-rays will in general be small com- pared to heavy particle emission when the latter is energetically possible because the coupling of the nucleus with the radiation held involves the small factor e 2 /hc = 1/137, DENSITY OF NUCLEAR ENERGY LEVELS- NUCLEAR TEMPERATURE The density of nuclear energy levels increases rapidly as a function of energy. To see how this comes about a model which is only a crude: approximation is used. We consider the nuclear 110 TOPICS NOT RELATED TO NUCLEAR FORCES particles as independent of each other, and suppose each of them has a set of equally spaced energy levels spaced by an energy difference A. Then, the excited states of the .system will also be spaced by the Interval A, and will have a greater statistical weight the greater the excitation energy, because of the greater number of ways of dividing the energy among the particles. When an interaction among the particles is then introduced, there will be splitting of each energy level ; and the statistical weight of an energy level of the non-interacting system is a measure of the energy level density in the same region of the spectrum, after the interaction has been introduced. To calculate the level density a model of the nucleus must be used. Four models will be mentioned. (For more details set: Nuclear Physics B, p. 79.) 1. Free Particles in a Box of the Size of the Nucleus. The level spacing D is proportional to exp( — V E), where E is the excitation energy of the nucleus. For A = 120, E = 3 Mev, we get D ~ 10 ev, which is about what is observed. 2. Free Particle in a Box, with Correlations. Bardeen has pointed out that the free particle model must be modified to be in accord with the assumption of exchange forces. The result gives a level spaaing depending on excitation energy in about the same way as before, but the level spacings are somewhat wdder: D ~ 100 ev for A = 120, E = 8 Mev. 3. Lattice Model. This model is the opposite extreme of models 1 and 2, for the particles are here supposed to be firmly bound and capable only of small vibrations about equilibrium. The re- sults are similar to those for models 1 and 2. The level spacing is proportional to expf-J^). For A = 120 and E = 8 Mev, t>~ 100 ev. 4. Liquid Drop Model For heavy nuclei this model is quite a good approximation. The level spacing is proportional to exp(-i? V; ) for small S and exp(-/i H ) for larger E. For A = 120 and E = 8 Mev, D — 10 ev. All these models give a level spacing which is a decreasing func- tion of the energy of the form expL— /(£)], where f(E) is a slowly variable function of the energy. If the density of states, p{E) = 1/D, of any system is given as a function of energy then an entropy can be defined as THE COMPOUND NUCLEUS 117 £ = k log p(E), and a temperature as dS/dE = \/T(E). Each of the four models mentioned will therefore define a nuclear tempera- ture lis a function of excitation energy. It turns out that for 10 Mev excitation energy, hT is of the order of 1 Mev, i.e., T = I0 10o K. The most satisfactory treatment of nuclear thermodynamics (Weisskopf, Phys. Rev. 52, 295, 1937) avoids a model and supposes I) = C exp(-L'Vi') (213) The constants B and C are determined from experiment: For low excitation energies the exponential is close to 1 so that D is about equal to C. From the observed position of the lowest excited levels, it is found that: For light nuclei (A ■ For heavy nuclei (A 20) -200) a- io H 10 3 (214) B can then be determined from neutron resonance levels near E rv 8 Mev (binding energy of neutron in nucleus); this gives about : B = 2 for light nuclei (215) B = 4 for heavy nuclei if E is measured in Mev. Any of the level density functions lead approximately to a Boltzmann distribution for inelastically scattered neutrons. If the incident energy of the neutrons is Bq and the energy of the emitted neutrons is IF then tin.: excitation energy of the residual nucleus is E n - W. Supposing that, the level density of the residual nucleus is exp [+/(£)] and expanding, m =/(e^ -rmw+ ■■■ (m w : e get a level density exp f(E) - exp f(E q ) X exp( -fW) (21 7) Therefore, setting /' = 1/lcT (which is exactly the expression demanded by dS/dE = l/T) gives a Boltzmann distribution for the level density of the residual nucleus as a function of W and therefore for the kinetic energies of the emitted neutrons. A more careful consideration gives a probability of emission proportional US TOPICS HOT RELATED TO NUCLEAR FORCES to VW «K P (-W/kT) or W ^ 9 ^-W/kT) but experiment lias not as yet given enough data to make it possible to distinguish between them. RESONANCE PHENOMENA Let the energv levels of a nucleus Z A be as shown in Fig. 10 and consider the process 7^ + i> - Z A U the incident neutron lias exactly the right energy to form Z A in one or its excited states the probability of capture is large. Such energies are called resonance energies of the compound nucleus. The experimental Vp- Resonances _Binc!ing energy ""'of neutron -^^.Ground state Fig. 16. Energy levels of a au&leus, Pig. 17. Typical experimental cross -seetiuii of a nucleus for slow neutrons- evidence (see Fig. 17) for neutron resonance energies m capture processes led to the first theories of the compound nucleus .ex- perimental ly, for A ~ 100, the level spacing D is about 10 ev, if E is about the binding energy of the neutron, i.e., 8 Mev. D is about the same at A ~ 200, and the appropriate binding energy E ~ 5 Mev Tins can be understood because, on the one hand, the number of particles is greater (and thus there are more possi- bilities of distributing the energy) ; on the other hand the excita- tion energy (binding energy of the particle) is smaller. For A smaller than 100, the level spacing increases rapidly. ' There are several nuclei for which m0 re than one resonance is knowm Among elements having only one (abundant) isotope, In has 3 resolved resonances, I lias 5, and Ta 7. In addition, many other elements show more resonances than isotopes. .Most of the experimental evidence was obtained by Rainwater, Havens, and their collaborators, in several papers in Phys. Rev. 71 £1947). In some; cases, onlv one resonance is observed; the level spacing is then not directly known but it pan be taken as of the same order of magnitude as the kinetic energy of the neutrons corresponding to the first resonance. THE COMPOUND NUCLEUS 11!) For protons, capture resonances have been observed only for the very light nuclei. The level spacings are of the order of 10 to 100 kev with an excitation energy of -10 Mev. lor heavier nuclei the Coulomb barrier prevents capture resonances for protons because the excitation energies which result after a proton lias been given sufficient energy to get over the Coulomb barrier are so high that the resonance levels overlap. A few resonances have also been observed for a-particles, the reactions of which lead mostly to the emission of protons or neutrons. The width T of a nuclear energy level is defined as Y = n, r, where r is the lifetime of the level. For most of the slow neutron capture levels the width is about 0.1 ev. This can be decomposed r=r 7 + r vl (218) into the neutron width and the y-ray width. Almost all of r is T which means that capture is far more probable than scattering for slow neutron resonances.* T, t may be determined separately in two different ways. First, the capture cross section at exact resonance is given by (const) X TjE r T (219) p is the width of the resonance at half-maximum; therefore, T\ can be determined from , at resonance, T and E T , Second the ratio of scattering to capture cross sections at resonance is iyi T , and f, is very nearly equal to i\ Unfortunately, in order to get the scattering cross section at resonance it must be ^entangled from the potential scattering (Nuclear Physics B, p. 162) so that this second method is ordinarily not ot much use. . The first experiments on neutron capture were done by fiermi and his collaborators, and by Moon and Tillman, using an ingen- ious but rather complicated method: a neutron beam from which the thermal neutrons had been removed by a cadmium absorber impinged on an indium detector. Comparison of the radmactivi- ties produced in this detector with and without an indium absorber intervening, showed that neutrons which activated the indium detector were strongly captured by the indium absorber It a silver absorber was used instead, the absorption was small. On * Mn has a Strong resonance at about 300 ev which gives mostly scattering ^"therefore has r n » iy TWs is lo be expected for light nuele: because of their large level spacing; see equation 206a.-Ae(e added m prwf. 120 TOPICS NOT RELATED TO NUCLEAR FORCES the other hand, a silver detector showed about as much radio- activity with and without the indium absorber, but with a silver absorber the beam was very strongly attenuated. The conclusion was that indium and silver were activated by neutrons of two different energies. At present, the most satisfactory method con- sists in using a modulated cyclotron beam and determining the velocity of the neutrons by their time of flight to tin- target. For very slow neutrons, a pile and a crystal spectrometer are often preferable. THE DISPERSION FORMULA Breit and Wigner were the first to develop a theory of nuclear resonance processes. The result was analogous to that in the theory of optical dispersion n - ■ 1 -j, (220) The measurements using velocity selection can check the shape of this curve and at the same time determine E T and P. To get the coefficient of proportionality in equation 220, suppose that the cross section cr is for the production of B with A incident. Then, since the cross section is proportional to the half-width for disinte- gration into B, it must contain fs, It also must contain T A for symmetry reasons. This follows from the principle of detailed balance: apart from statistical weights and a factor depending on the ratios of momenta, o- A _> B should be equal to ff B -,A- (See Chapter XL) Finally we know that for the simplest case in which only one kind of particle can be emitted or absorbed, T A = Fs = F, and we know further that in this instance the largest possible cross section for particles with I = is IttX 2 . Clearly, in the general case, the wave length of the incident particle must occur. Collect- ing all information, a = %X, 2 Va Tu (221) (S - E r f + (l'/2) 3 This is known as the one-level Breit- Wigner formula. It gives the correct dependence on momentum, in accord with the prin- ciple of detailed balance THE COMPOUND NUCLEUS 121. For the dependence on the spin of the compound nucleus and the generalization to more than one resonance level see Nuclear Physics B, p. 101. There is only one instance in which the many- level formula has been of use, namely, He 4 + n -* He s -» He 4 + ft (223) which has two partly overlapping resonances near 1 Mev. The dispersion formula has been derived many times. The derivation must be quite different from, the treatment in optics, where the interaction of the incident light and the atom can be taken as a small perturbation. For high-energy neutrons the dispersion theory goes over into the statistical theory given previously. The partial widths of the levels become of the order of magnitude of the level spacing and the resonances are no longer observable. For extremely slow neutrons, well below the first resonance, Ta. is proportional to v (this follows from the fact that F A is propor- tional to the density of states in momentum space, j/{dp/dE) rv -p) and so the Breit-Wigner formula reduces to X 2 v rv 1/s (224) This is the well-known .1 fv law for the cross section at very low energy. It makes the number of processes per second, which is av, independent of the energy distribution and proportional only to the total particle: density. For very light nuclei, the spacing D is very large and the 1/v law holds up considerable energies. For gin + ?l ^B" it is valid to 50,000 ev. Absorption by B 10 is therefore used for measuring neutron velocities. APPENDIX: TABLE OF NUCLEAR SPECIES Explanation" of the Table Column 1: Column 2: Column 3: Column 4: Column 5: K e~ Z." Atomic number of the element. Element." Chemical symbol of element. A." Mass number of the Isotope. Abund., per cent." Per cent abundance of isotope in the afurally occurring element. Disintegration." Symbols for nuclear processes are: 1 isomeric transition. (Emission of ->-rays or conversion electrons.) Electron capture. Internal conversion electrons. Negative, positive beta-particle emission. a Alplia-particle emission. n, H Emission of neutrons, protons. V Denotes that the paitieular isotope has not been identi- fied with complete certainty. Parentheses enclosing one or more activities denote uncertainty in these, but not in the identification of the isotope to which they are assigned. Thus, 4; Ag 103 has been classified and found definitely to have fi" activity; however, it is not certain that 47 Ag 108 also has K-capture and conversion .elec- trons. A comma setting off le" from one or more symbols indicates that the conversion electrons belong to the isomeric transition. Columns and 7: Masses, with probable errors. A value in parentheses indicates that the mass has been estimated from theory, the isotope not having been produced as yet. Column 8: Spin of the designated isotope. Main Reffrenous G. T, Scaborg, Table of Isotopes, Rev. Modern Phys. 16, I, 1944. E. Segre, Isotope Chart, issued by Los Alamos Scientific Laboratory, 1 9*8, In general, isotopes classified as A to D by Seaborg and Segre have been included in this table, i.e., all those for which at least the assignment to a definite element is certain. 123 121 Abund., APPEN Disinte- DIX Error Z Element, A per eeat gration Mass X UJ 6 Spin n 1 1.008 93 3 H 1 H 1 99.98 1.008 123 O.fi H 2 . 02 2.014 70S 1.1 i 3 (T 3.017 02 3.4 H 2 He a ~10~ 5 3.017 00 4 4 100 4.003 90 3 5 n 5.013 7 35 r 6.020 9 50 3 Li 5 ii (5.013 6) 60 6 7.5 0.016 97 5 l 7 92 . 5 7.018 22 6 n 8 f 8.025 02 7 4 Br () (6.021 9) 100 7 K 7.019 16 7 8 tx 8.007 85 7 9 100 9.015 03 6 % 10 p- 10.010 77 8 11 (11.027 7) 5 B 9 9.016 20 7 10 18.4 lO.Olfi 18 9 i 11 81.6 11 .012 84 8 H 12 r 12.019 70 13 (13.020 7} 6 C 10 (? + 10.021 30 11 [i+ 11.014 95 9 12 98.9 12.003 82 4 IS 1.1 13.007 51 K) H 14 p- 14.007 67 5 15 (15.016 5) 7 N" 12 (12.023 3) 13 + 13. 009 88 7 U 99 . 62 14.007 51 4 i 15 0.3S 15,004 89 21 H Hi r > 16. 006 5 < 16.011 17 (17.014) 8 14 (14.013 1} 15 v + 15.007 8 40 IS 99.757 16.000 000 Standard 17 0.039 17.004 50 a IS 0,204 IS. 004 9 40 19 P~ 19.013 9 TABLE OF NUCLEAR 3PECIE8 Abund., Disinte- Error z Element A per cent gration Mass X 10* 9 F 16 (16.017 5) 17 ff* 17.007 S 30 18 .3+ 18.006 5 00 19 100 19.004 50 26 20 r > 20. 004 2 <20.009 2 21 (21.005 9) 10 No 18 (18.011 4) 19 + 19.007 81. 20 20 90.00 19.998 77 10 21 0.27 20.999 03 22 22 9.73 21.998 44 30 23 r 23.001 3 11 Na 21 p + 21.003 5 22 $+ 21.999 9 50 23 100 22.996 18 31 24 a~ 23.997 5 45 25 Ufr (24.996 7) 12 Mg 22 (22.006 2) 23 ft + 23.000 2 40 24 77.4 23.992 5 00 25 11.5 24.993 8 90 26 11.1 25.9S9 8 50 27 r 26.992 8 150 13 Al 25 3 + 24.998 1 100 20 P + 25.992 9 150 27 100 26.989 9 80 28 &~ 27.990 3 70 29 P~ 28.989 3 80 30 (29.995 4) 1.4 Si 27 P + 26.994 9 90 28 89.6 27.986 8 60 29 6.2 28.986 6 70 30 4.2 29.983 2 90 31 P~ 30.986 2 (iO 32 (31.964 9) 15 P 29 e+ 28.991 9 100 30 p + 29.9S7 3 10 31 100 30.984 3 50 32 r 31.9S2 7 40 33 (32.982 6) 34 $- 33.9S2 6 40 125 Spir H ; 126 APPENDIX A bund., DltfMLilV Error Z Element A per cent gration Mass X 10 s Spin m S 31 p + 30.989 9 32 95.1 31. 980 89 7 33 0.74 32. 980 GO 34 4.2 33. 977 10 33 35 F 31. 978 8 SO 38 0.016 35. 97S 100 37 §~ 36. 982 1 30 17 CI 33 # 32 986 34 P 33 980 1 200 35 75.4 34 978 67 21 H 36 <3 + ,S" K 35 97S 8 100 37 24.0 36 977 50 11 % 38 fr 37 981 300 39 (3S 979 4.) 18 A 35 r 3-1 985 36 0.307 35 978 100 37 K 36 977 7 38 0.061 37 974 250 3D (38 975 5} 4!) 99.632 39 975 6 60 41 r 10 977 60 19 K 37 (36.983 0) 38 3 + 37 .979 5 39 93.38 38 .974 7 % 40 0.01.2 e-x 39 .976 100 l 41 6.61 10 .974 % 42 w ■13 Ufr 20 Ca 39 VP* 40 96.96 39 .975 3 150 (.) 41 TJ K g- 42 0.64 41 .971 1 43 0.15 ■12 .972 3 ■11 2.06 ■15 r 46 0.0033 48 0.19 49 r 21 Bo 41 43 It 45 100 44 .966 9 so % 46 S~ if Z Element 21 Sc {cant.) 22 Ti 23 V 24 Cr 25 Mn 26 Ee 27 Co TABLE OF NUCLEAR SPECIES A 47 48 49 15 46 47 48 49 50 51 47 48 49 50 51 52 49 50 51 52 S3 54 55 51 52 54 55 56 53 54 55 56 57 58 59 55 56 57 58 59 60 Abund,, per cent 7.95 7.75 73.45 5.51 5.34 100 4.49 83.78 9.43 2.30 100 0.04 91.57 2.11 0.28 Disinte- gration r (3+ U ,3 '" £+ K a k r U K cT V f3-'~K K 100 a+ k $+ K r 13+ K Mass 45.966 1 46.964 7 47.963 1 48.964 6 49.952 1 50.958 7 50.957 7 50.958 51.956 52.956 54.957 53.957 Error X 10 5 127 Spin 100 100 50 60 40 100 50 % 55.956 8 170 56 . 957 n 128 ArPENDIX Abund., Disinte- Error '/, EI c men t, A per cent gration Mass X 10 & 28 M 57 & 58 67.1 57.959 4 10 59 V .a ' GO 26.7 59.949 S 40 61 1.2 60,953 7 150 62 3,g 61.949 3 40 ea r 0-1 0.88 63.947 1. 60 29 Cu 58 BO 01 62 e+ 0+ K P + 63 70.1,3 62.957 400 G4 r& K 56 29.87 64.955 400 66 r 30 Zn 63 ^ 64 50 . '.) 63.955 400 65 fl • K f m 27.3 65.954 400 67 3.9 66.954 400 68 17.4 67.955 300 69 li~ I 70 . 5 69.954 300 31 Ga (it 65 m 67 68 U 3+ K '~e~ fi + K ,.r 69 61.2 68 . 952 800 70 8~ 71 38.8 70.952 900 72 0~ 71 V fi- 12 Ge 69 70 71 72 73 74 75 76 77 78 21.2 27.3 7.9 37.1 fi.5 ll 3+ A* ( - «r Spin H % % TABLE OF NUCLEAR SPECIES Mass AbuTiJ., Disinte- z Element A per cent gration 33 As 72 73 74 75 76 77 78 100 U 3+ V P + K e r ? + &- /3 + K Up~ 34 He 74 75 76 77 78 79 80 82 83 0.9 9.5 8.3 2-1.0 48.0 9.3 K e" 35 Br 78 79 80 SI 82 83 84 85 87 50-6 49.4 3~, 1 <T r r r 129 Error X 10 5 Spin % M m Kr 78 79 80 81 82 0.35 2 01 1 I . 53 (./ 1 <~ 83 ] 1 . 53 i «r % 84 57.11 85 p~ 86 17.47 87 u tr 88 0- 89 ,a- 90 U fi- * 91 ll fr 92 ur 94 U fi- 95 ll 07 ^EJ ' J^™ 130 Abund., APPENDIX Disinte- Z Element A per cent gration 37 Rb 82 84 85 86 72.8 V U 0~ 87 27.2 P~ 88 tr gg r 'JO V fi- 91 ll rr 92 u&- !)4 U ii~ 95 v$r 38 gt 84 85 8(5 0.50 9.86 J <r K 87 7.02 le- 88 82.50 89 ft" 90 UfT 01 U fi- 92 ll fi- 01 ll ft- 95 ur 39 Y 86 87 88 89 90 91 92 04 95 100 UK (l e~) K V fi~, t e- U fi- ll fi- ll r 40 Zr 89 90 91 92 03 94 95 48.0 11.5 22.0 17.0 fi~, I or B U0- v a~ ■11 Cb 96 07 90 91 92 Mass Error X 10* Spin % TABLE OF NUCLEAR SPECIES ur U8+ UK.tr W A bund., Disinte- Error Z Element A per tent gration Mass X 10 5 41 Cb 93 100 (sent.) 94 95 96 97 a U 3~, I cr U 42 Mo 92 03 14.9 U (3+ 94 9,4 93.945 800 95 16.1 94.946 800 00 J6.G 95.944 800 97 9 . 05 96 . 945 900 08 24.1 97.943 000 09 v r 100 9.25 101 u r 102 U fi- 43 Te 96 98 99 101 102 ll K U K <T 6", I e~ U fi- ll r 41 Ru 90 5.08 95.945 1100 08 2.22 97.943 1100 00 12.81 98 . 944 1100 100 12.70 99.942 1100 101 16.98 100.946 1100 102 31.34 101.941 1100 103 r 104 18.27 105 fT 106 u r 107 O $r 45 Rh 102 s-ft + 103 100 I 102.941 1100 104 li~, lf~ 105 $r 106 U fir 107 rye- 46 Pd 102 0.8 101.941 1100 104 0.3 103.941 1100 105 22.6 104.942 900 106 27.2 105.941 1000 108 26.8 107.941 1000 131 Spin % H 132 APPEXDIX Abund ., Ksiiite- Error Z Element A per ecu t grat ion Mnss X 10 s Spin 46 Pd 109 u tr (cont.) 110 111 112 13,5 pr 109.941 1000 ■17 Ag 105 106 U K 3+ K B~ 107 SI. 'J {/O 106.945 600 ] A 108 r (K E ") 109 48.1 108.944 700 k no 8~ in P~ 112 w~ m Cd 106 107 108 110 111 112 113 114 IIS 116 117 1.18 1.4 1.0 12. S 13.0 24.2 12.3 2S.0 7.3 UK U, I ,- 49 In no .1 11 112 113 114 1.15 no 11.7 4.5 95.5 U 8+ V a+ «- U, I e~, I er 1 e~, 3~ 1 <r a~ Ke~ 50 Sn 112 11.3 114 1.1 O.S KtT 115 0.4 114.940 1400 H 116 15.5 115.939 1400 11.7 9.1 ne. 937 1400 H US 22.5 117.937 1400 110 9.8 118.938 1400 H 120 28.5 119.937 1400 121 U-ftr 122 5.5 121.945 1400 123 U8- TABLE OF NUCLEAR SPECIES Abund., Disinl.c:- Error z Element A per cent gration Mass X 10 5 50 Si) 124 6 8 123.944 1400 {cmd.) 125 127 128 if fir U 3~ 11 3- 51 m 120 121 122 123 124 126 127 128 129 132 133 136 56 44 ti- ll fir &~ &~ U3~ u 3- 52 Te 120 121 Q 088 IK<T) 122 2 43 (ier) 123 85 124 4 59 125 6 93 126 18 71 127 0~, 1 e~ 128 31 86 129 8~, I <r 130 34 52 131 8~, I e~ 132 tiff" 133 u 3- 135 fi~ tae u tr 137 , u a~ 53 1 124 1 26 127 128 130 131 132 133 135 136 137 100 3+ a~ K a- a- 3- Vb- V 8- u 3- 133 Spin % J 4 APP1 Abund., 2NDIX Disinte- Error z Element A per cent gration Mass X 10 s Bpin 54 Xe 124 126 127 128 1 094 088 90 U, I e~ 129 26 23 u 130 4.07 131 21 17 H 132 26 96 133 &)lr 134 10 54 135 (Tl 136 8 95 137 U P~ 138 Up~ 139 sr 140 r 141 0- 143 u tr 144 Up~ 55 Qa 130 132 133 134 136 137 138 139 140 141 142 143 100 V v k <r Uff- u p- pr Up- P~ v tr u % 50 Ba 130 132 133 134 135 •2 8 101 097 42 59 I er % 136 7 81 137 11 32 % 138 71 .66 G 139 p~ 140 p~ 141 p- 142 ur 143 ur 145 up- TABLE OF NUCLEAR SPECIES 135 Abund., Disinte- Error z Element A per cent gration Mass X 10 5 Spin 57 L:i 137 U K 139 100 138.953 800 K 140 r 141 P~ 143 U 3~ 144 up- 145 Up~ 58 Ce 136 138 140 141 142 143 L44 145 147 <1 <1 89 11 r Up~ u p- u p- U p~ 59 Pr 140 [41 142 143 144 145 147 100 p + p~ u r up- Up~ UfC M 60 Nd 141 142 143 144 25.95 13.0 22.6 p + 145 9.2 144.962 400 146 16.5 145.962 400 148 6.8 147.962 400 150 5.95 149.964 400 61 61 143 144 145 146 147 Up~ U I or K Up~ u up- 62 Sm 144 146 147 1.48 149 3 16.1 14. 2 15.5 VI a 136 A PPEXDIX Abund., Disinte- 2 Elemenl A per cent gration 62 Sm 150 11.6 (cont.) 151 u-fir 152 20.7 154 18.!.) S3 Eh 151 152 153 154 155 156 157 158 49.1 50.9 U fi~ e~ Utf V ti- ll ti- ll $- 64 Gd 152 154 155 156 157 158 160 0.2 1.5 18.4 19.9 18,9 20.9 20.2 66 Tb 159 160 100 fr OG Dy 158 160 161 162 163 164 165 XI. 1 0.1 21.1 26.6 24.8 27.3 it 67 Ho 165 166 100 u &- 88 Er 162 164 166 167 168 166 170 0.1 1,5 32.9 24.4 26.9 14.2 USi~ 6ft Tm 169 170 100 03-) Mass Error X IB 6 Spin m TABLE OF XUCLEAR SPECIES Mass 153.971 600 154.971 600 155.972 600 156.973 600 157.073 600 159.971 600 % H Abund., Disinte- z Element A per cent gration 70 Yb 168 170 17! 172 173 174 175 176 0.08 4.21 14.26 21.49 17.02 29.58 13.38 U 71 Lu 175 97.5 176 2.5 0- K 177 if ft 72 Hf 174 176 177 178 179 180 181 0.18 5.30 18.17 27.10 13.85 35.11 li~ 73 Til 180 (.3") IS. e 181 100 I 182 ft 74 W 180 182 183 184 185 186 187 —0.2 22 . 6 17.3 30.1 29.8 u p- w'fr 75 Re 184 185 186 1S7 188 38.2 61.8 UK U P~ u p- 76 Os 184 186 0.018 1.59 187 1.64 K 188 13.3 189 16 . 1 190 26.4 191 v p- 192 41.0 193 Ujr Error X 10 s 137 Spin X >7 <% <H 7 A 3k % 189.04 2000 Viar% 190.03 2000 192.04 2000 13S AI Abund., PENDIX Ilisinlr- Error Z Element A per cent graf.ion Mass X 10 s Spin 77 Ir 191 192 3S.5 r 191 .04 2000 H 193 61. 5 193.04 2000 % 194 P~ 78 Pt 192 0.8 194 30.2 194.039 1400 195 35.3 195.039 1400 M 198 26.6 (/O 196.039 1400 197 jjf 198 7.2 198.05 2000 199 r 79 An 196 TJ $T e~ 197 100 I 197.04 1000 % 198 §r e~ 199 l3~ 200 v&- SO Sg 190 197 198 0.15 10.1 Ke~ 199 17.0 I er H 200 23.3 201 13.2 U 202 29.6 203 ff'tr 204 0.7 205 ir 81 Tl 198 199 202 U K f V K "e~ U K t~ 203 29.1 203-05 2000 H 204 Upr 205 70.9 205 . 05 2000 % 206 ur AcC" 207 r ThC 208 r Tl 209 p- RaC 210 e- 82 Pfa 203 u $+ 204 1.5 204.05 2000 205 v, i r 206 23.6 206.05 2000 207 22.6 207.05 2000 }£ TABLE OF NUCLEAR SPECIES Abi ind., 11 i si ii fce- Error z Element A per eent gration Mass X to? 82 Pb 208 52.3 208.05 2000 icont.) 209 8~ RaJD 210 tt AcB 21] 0- ThB 212 g" Pb 213 fr SaB 214 r S3 Bi 207 K sr 209 100 209.05 2000 RaE 210 i'~ AcC 211 ft" a ThC 212 §T\M, Bi 213 p~ a Rati 214 f}" a 84 Po 210 a AcC 211 a ThC 212 a Po 213 a RaC 214 a AoA 215 <x ThA 210 (i~a Po 217 a RaA 218 r a 85 At 21 I K a 88 An 219 & Tn 220 a Rn 221 a Rn 222 a 87 87(AeK) 22;; v pr 88 AcX 223 a ThX 224 a. Ra 225 a Ra 226 a MsTbi 228 v~ m Ac 227 fi- <z MsTh 2 228 l3~ a 90 RdAo 227 a RdTh 228 a Th 229 a To 230 a UY 231 $r 139 :pm 140 APPENDIX Ab unci., Disinte- Error Z Element A per cent gration Mass X io 5 Spin 90 Th 232 100 a 232.11 3000 (cord.) Th 233 r UXi 234 &" 91 Pa 231 232 233 a {j- - X m 234 r VX-i 234 wi 92 U 233 a UII 234 0.00518 a AcU 235 0.719 a u 237 r UI 238 99.274 a 238.12 3000 u 23'.) r 93 Np 234 235 256 237 238 239 K K r ix ST r 94 Pii 238 239 a a 95 Am 241 a 96 Cm 240 242 ® ^ INDEX Allen, 22 Allowed transitions ( h <?-decay), 97 in intermediately heavy nuclei, 104 in mirror nuclei, 103 in naturally radioactive nuclei, 104 lifetimes in, 102 a-parlicle, rote of, 84 m-parlicle emission, 14 a-r&d inactivity, lifetimes for, 6 Alternation, intensity, in band spec- tra, 18 Alvarez, 24, 53, 54 Amaldi, 91, 93, 115 Angular distribution, of neutron- proton scattering, 39 at high energy, 03 of proton-proton scattering, 69 Angular momentum, 38 Annihilation, of electrons and posi- trons, 2 of heavy particles, 2 Arnold, 25, 28 Atomic weight, 3 Atoms, theory of, 23 Axial vector, 107 Backer, 115 Backus, 102 Bailey, 44 Band spectra, 15 Barrier, potential, 7 penetration of, 115 Bartlett force, 82 Beams, molecular, 15 Bennett, 44 Benjstralh, 44 Be 8 , 5 ^disintegration, 10, 97 and neutrino, 20 Fermi theory of, 97 ^-disintegration, Konopiriski-U hi en- beck theory of, 100 Kurie plot of, 100 of neutron, 105 selection rules in, 99 ^-lifetimes, 102 /3-spectrum, 20 Bcihc, 105 Binding energy, of deuteron, 29 of neutron in heavy nuclei, 1 1 7 Binding of proton in molecule, effect of, 47 Block, 15,24 Bohm., 44 Born, 37 Bom's approximation, 18, 109 Bose statistics, 16 Breit, 69, 70, 71, 120 Breit-Wigrier formula, 1 20 Bretscher, 60 Brickwe.dde, 53 Britlouirt, 7 Erode, 94 Bulk matter, internal energies of, SO Capture of neutrons, by protons, 00, 79 in heavy nuclei, 1 18 Cenler-of-mass coordinates, 37 Chadwich, 29, 60 Champion, 92 Charge, I Charged meson theory, 88, 95 Charged particles, nuclear reactions involving, 8 Chemical bond effect on neutron scattering, 47 Chemical properties, 1 Compound nucleus, 109 disintegration of, 1 12 141 INDEX 143 142 INDEX ' fl I 1 1! Ill , ill Compound n&ofcja, formation of, 111 lifetime of, 110 Compton wave length, 94 Condon, Conservatism of energy, 5 Constituents of nuclei, 8 Cork, 100 Coulomb field, 101 Coulomb repulsion, 6, 8 Coulomb scattering, 64, 65 Critchjidd, 105 Cross section, for nuclear reactions, 8, 114 for capture of neutrons by protons, B0, 79 for photoelectric disintegration of deuteron, 56, 79 for scattering, of neutrons by heavy nuclei, 7, 114 of neutrons by protons, 37, 40, 79 of protons by protons, 64 geometrical, 7 for capture by heavy nuclei, 111 total, 40 D-D neutrons, 42 de Brogiie wave length, 8 of electrons, 9 of neutron or proton, 8 do Brogiie wave number, 37 Decay, £-, see ^-disintegration Decay processes, 13 Dee, 40 Density of nuclear energy levels, 115 Depth of nuclear potential well, 32, 70 Determination of force constants, 70 Deuteron, excited slates of, 34 ground state of, 29 interaction of, with radiation, 56 magnetic moment of, 26 photo disintegration of, 79 physical properties of, 25 quadrupole moment of, 27 singlet state of, 43 states of, with tensor forces, 74 virtual state of, 45 wave function of, 33 DeWirc, 53 Dipole radiation, 13 Dirac S-function, 95 Disintegration, p, 10, 97 nuclear, 13 of compound nucleus, 112 D i sin I .e gra li on probabi 1 i I i es , 1 1 3 Dispersion formula, 120 Dispersion theory, 121 Distribution, angular, at high energy, 63 Dunning, 53 Einstein's relation, 2 Eisenbud, 69, 70, 87 Electric dipole moments in nuclei, absence of, 75 Electron energies, distribution of, in ^-disintegration, 18 Electrons, annihilation of, 2 in nucleus, 5, 9 spontaneous emission of, 1.0 Electrostatic interaction of protons, 7 Emission, of a-particles, 14 of /3-rays, 10, 14 of 7-rays, 13 of heavy particles, 13 of light quanta, 14 of neutrons, 13 of protons, 14 Energy, 5 conserved, 5 kinetic, 5 Energy distribution in inelastic scat- tering, 114 Energy equivalent, 2 Energy levels, nuclear, density of, 115 Exchange and spin, relation between, 84 Exchange forces, 81 and saturation, 83 effects of, 82 quantitative theory of, 87 Excited states, of deuteron, 34 of nucleus, 111 Experiments, on neutron-proton scat- tering, 42, 91 on photo disintegration, 60 on scattering by para-hydrogen, 49 Experiments, proton-proton, advan- tages of, 64 East neutrons, 7 Fast nuclear particles, uncertainty in position of, 111 Feather, 60 Fermi, 43 Eermi interaction, 97 Fermi statistics, 16 Fisk, 81 Forbidden transitions, half-lives, 106 Force, Bartlett, 82 Heisenberg, 82 Majorana, 82 Wigner, 82 Force constants, determination of, 76 Forces, exchange, 81 quantitative theory of, 87 non-central, 73 nuclear, meson theory of, 94 ordinary, B on-sat urati on of, 81 prol.on-prol'.on, 64 saturation of, 80 short-range, 66 Formula, dispersion, 1 20 Free particle model, 116 Frixch, 44 Frostier, 94 Fundamental particles in nucleus, G 7-rays, emission of, 13 Gamow, 6, 106 Gamow-Teller selection rules, 106 Geometrioal cross section, 7, 111 Gurlach, 15 Gilbert, 40 Go'ppert^Mayer, 33 Goldhaber, 29, 60 Gordon, 94 Graham, 60, 111 Ground state, of deuteron, 29 of He 2 , (54 Gurney, 6 Gyromagnetic ratio, 24 Hafslad, 66 Halban, 60 Ilamermesh, 53 Hansen, 15 H ovens, 118 Haxby, 20 Heist'ttberg, 81 Heisenberg force, 82 H eider, 56 He-\ 5 Herh, 70 Heydenburg, 66 High-energy neutrons, experiments with, 91 High-energy phenomena, information from, 62 Hyperfine structure of spectra, 15 Inelastic scattering, energy distribu- tion in, 114 Information obtainable from high- energy experiment, 63 Intensity alternation in band spectra, 18 Interaction, electrostatic, of protons, 7 of deuteron with radiation, 56 Interference between nuclear and Coulomb scattering, 64 Invariance, against inversion, 73 relalivistie, 107 Inversion of coordinate system, 73 Isobars, 2, 1 1 of neighboring Z, 1.1 stability of , 1 1 "Isomer of nucleus, 13 Isotopes, defined, 1 Isotopic spin, 84 Isotopic spin functions, 85 Jentschke, 29 K electron capture Kawne, 71 Kellogg, 23 Kemhle, 74 Keener, 78 Kerst, 70 Kikuchi, 101 Kimura, 29 Kinetic energy, 5 10, 14, 22, 108 115 mr 144 INDEX Klei a -Gordon equal, ion, 94 Kmuypinski, 97, 103 Konopinsld-Uhlenbeek theory, 100, 104 Kramers, 7 Krugcr, 92 Kurie plot, 100 Ladenbnrg, 40 Lande's formula, 28 Lmmhlin, 92 Lawson, 100 Levels, n u clear energy, density of, 115 width, experimental, 119 Lifetimes, p-, 102 in allowed transitions, 102 in forbidden transitions, 105 LigM quanta, emission of, 14, 119 Liquid-drop model, 80 Li 5 , 5 Low-energy phenomena, information from, 62 Lyman, 101 Magnetic moments, 15 of deuteroii, 25 Majorana forte, 82 iliti.j7i.fle/er, 29 Mass, reduced, 37 M.ass excess, 4 Mass number, I Mass spectrograph, 3 Mosses, 37, 66, 67 Matter, bulk, internal energies of, 80 Meson theory of nuclear forces, 88, 94 Metastable state of nucleus, 13 Molecular beams, 15 Molecular velocity effect on neutron scattering, 49 Moment, magnetic, IS quadrupole, of deuteron, 27 Moon, 1 19 Morse, 81 Mott, 37, 66, 67 Myers, 29 Neutral meson theory, 88 Neutrino, 20 Neutrino, /3-disintcg ration and, 20 experimental evidence for, 21 Neutrino mass, 20 Neutron, p-decay of, 105. physical properties of, 24 slow, cross section of, 43 wave length of, 7 Neutron emission, 13 Neutron-proton scattering, 78 experimental results on, 42, 69 at high energies, 91 Neutron spin, 45 evidence for, 45 from scattering, 45 Neutrons, capture of, by protons, 60, 79 Li-D, 42 fast, 7 in nucleus, 5, 8 scattering of, by ortho-hydrogen, 49 by para-hydrogen, 49 by protons, 37, 78 at high energies, 91 by protons bound in molecules, 47 thermal, 43 Non-central forces, 73 Non-saturation of ordinary forces, 81 Non-zero spin, nuclei of, 18 Nuclear abundance, 12 Nuclear charge, 1 Nuclear constituents, 15 Nuclear disintegration, 13 Nuclear energy levels, density of, 1 15 width, 119 Nuclear force, between neutron and neutron, IS, 71 between neutron and proton, 30 between proton and proton, 64, 71 meson theory of, 94 saturation of, 80 spin dependence of, 45 Nuclear reactions, cross section for, 114 general theory of, 110, 114 involving charged particles, 8, 115 INDEX 145 Nuclear scattering, 7, 64, 67, 114 interference between Coulomb scat- tering and, 64 Nuclear species, table of, 123 Nuclear spin, 15, 23 Nuclear temperature, 115 Nucleus, absence of electric dipole moments in, 75 basic facts on, 1 oompouTirl, L09 disintegration of, H2 formation of, 111 lifetime of, 110 constituents of, S excited states of, lit fundamental particles in, 5 isomer of, 13 magnetic moment of, 15, 25 metas table state of , 13 quadrupole moment of, 27 residual, 111 size of, stable, regularities in, 2 Nuckolls, 44 Number, mass, 1 Octopole radiation, 13 Grtho-hydrogen, scattering of neu- trons by, 49 Packing fraction, 4 Para-hydrogen, scattering of neu- trons by, 49 Parity of wave function, 75 Parkinson, 70 Partial width of level, 1 1 4 Particles, charged, nuclear reactions involving, 8 heavy, annihilation of, 2 emission of, 13 Pauli principle with isotopic spin, 86 Peierls, 78 Penetration of potential barrier, 115 Phase slrifts, 37 for I ?s 0, 38 Photodisintegration, 56, 79 experiments on, 60 Photoelectric effect, 56 Photo magnetic effect, 57 Physical properties, of deuteron, 25 of neutron, 24 of proton, 23 Titter, 53, 54 Plain, 70 Polar four vector, 107 Position of fast- nuclear particles, un- certainty in, 11 1 Positrons, annihilation of, 2 spontaneous emission of, 10 Potential, relation between range and depth of, 32, 70 Potential barrier, penetration of, 115 Potential well, rectangular, between two protons, 70 of deuteron, 31 Powell, 92 Probability, disintegration, 113 penetration, 115 sticking, 111 Proof of saturation, 84 Properties, chemical, 1 Proton, physical properties o) , 23 Proton emission, 14 Proton-proton e^Srka^tttS, advan- tages of, 64 Proton-proton forces, 64 Proton-proton scattering, 64 experiments on, 70 theory of, 65 Proton resonances, 119 Protons, bound in molecules, scatter- ing of neutrons by, 47 capture of neutrons by, 60, 79 in nucleus, 5, 8 scattering of, by protons, 64 scattering of neutrons by, 37 Pseu do scalar, 107 Purcell, 15 1 Quadrupole moment, 27 Quadrupole radiation, 13 Quantitative theory, of exchange forces, 87 of nuclear forces, 23 Quantum mechanics, 23 14S INDEX Rabi, 15, 23, 27 Radiation, dipote, 13 interaction of deuteron with, 56 ootopole, 13 quadrupole, 13 Radioactivity, a, lifetimes for, 6 if, 10, 97 Ragan, 71 Rainwater, 1 IS Ramsey, 23 Range, of nuclear forces, 32, 49, 70 ia relation to depth, 32 Rari.ta, 27, 59, SO, 73, 76, 77, 87, 88, 89 Reactions, nuclear, involving charged particles, 8 Recoil energy, nuclear, 21 Rectangular potential well, 31, 70 Reduced mass, 37 Regularities in stable nuelei, 2 Relation, between range and depth of potential, 32 between spin and exchange, 84 Relativistic in variance, 107 Residual nuclei, 111 Resonance phenomena, 118 Resonances, proton, 119 Richards, 44 R:idiardson, 101 Rich-man, 44 Roberts, 26, 28 Role of ^-particle, 84 Rotation of coordinate system, 73 Rules, selection, Fermi, 99 Gamow-Teller, 106 Rutherford, 65 Sachs, 33 Saturation, of nuclear forces, 80 proof of, 84 Seaixrrg, 111, 123 Segre, 123 Scalar in /5-theory, 107 Scattering, Coulomb, fl'l interference between nuclear and, 01 inelastic, energy distribution in, 1 1 4 nuclear', 64, 114 Scattering, of neutrons, by heavy nu- clei, 7, 1 14 by ortho-hydrogen, 49 by para-hydrogen, 49 by protons, 37, 78 angular distribution of, at high energy, 63 - experimental results on, 43, 91 spherical symmetry of, 39 total cross section for, 40 by protons bound in molecules, 47 of protons by protons, 64 angular distribution of, 69 shadow, 114 spherical symmetry of, 39 Scherrer, 102, 108 Schdff, 81 Schrodinger equation, 31 Schwinger, 27, 28, 49, 50, 53, 59, 60, 73, 76, 77, 87, 88, 89 Selection rules, Fermi, 99 in /3-dccay, 99 Gamow-Teller, 106 Shadow scattering, 114 Short range of nuclear force, 30, 60 Skoupp, 20 Singlet state, of deuteron, 43 sign of energy in, 45 Size of nucleus, 6 Slow neutron cross section, 43 Spectra, band, 15 Spectral lines, splitting of, 15 Spin, 15 and exchange, relation between, 84 and isotopic spin, 84 and statistics, 15 iso topic, 86 nuclear, 15 Spin dependence of nuclear force, 45 Splitting of spectral hues, 15 Spontaneousemission,offf-particles,6 of electrons, 10 of positrons, 10 Spherical symmetry of scattering, 39 Stability, of isobars, 11 of nuclei, 5 INDEX 147 Stable nuclei, regularities in, 2 States, excited, of nucleus, 111 of deuteron, 74 Statistical considerations, 60 Statistical weights, 01 Statistics, 10 Bose, 16 Fermi, 16 of neutrons, 1 S of protons, 18 spin and, 15 Staub, 40 Stern-Gcrlach experiment, 15 Stetier, 29 Stevens, 20, 30 Sticking probability, 111. Structure of spectra, hypcrfine, 15 Survey of low-energy phenomena of deuteron, 62 Sutton, 53 Symmetric meson theory, 88 Symmetry of wave funciton, 68 Tashelc, 71 Teller, 49, 50, 106 Temperature, nuclear, 1 15 Tensor in /3-theory, 107 Tensor forces, 28, 75 Thaxton, 69, 70 Theory, of atoms, 23 of exchange forces, 87 Thermal neutrons, 43 Thomas, 31 Tillman, 1 19 Total cross section for scattering of neutrons by protons, 40 Townsend, 101 Transitions, allowed, in /3-d i si nteg ra- ti on, 99 lifetimes in, 102 forbidden, lifetimes in, 105 7-ra.y, 13 Transmission coefficient of barrier, 7 Time, 66 Tyler, 100 UMeribeck, 97 Uncertainty in position, 111 l/v law, 121 Van Attn, 29 Van de Graaff machine, 29 Variation principle, 80 Virtual state of deuteron, 15 VolkajJ, 82 Wave function, of deuteron, 33 parity of, 75 symmetry of, 68 Wave functions, relativistie eom- bi nations of, 107 Wave length of neutron, 7 Wave number, de Broglie, 37 Wei glits, statistical, 61 Wei.sskopf, 113, 117 Wells, 20 Wentz el -B rilloui n-Kramers method, 7 White, 66 Wick, 93 Width of level, 113 partial, 1 14 Wiendenbeck, 29 Wianer, 28, 30, 43, 44, 45, 52, 73, S4, 87, 120 Wigucr argument about short-range forces, 30, 66 Wigner forces, 81 Williams, 44 Wilson, 70, 92 Yukawa, 94 Zaeharias, 23 *